THE GEOMETRY OF OPTIMAL AND NEAR-OPTIMAL

RIESZ ENERGY CONFIGURATIONS

By

Justin Fitzpatrick

Dissertation

Submitted to the Faculty of the

Graduate School of Vanderbilt University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

in

Mathematics

August, 2010

Nashville, Tennessee

Approved:

Professor Douglas P. Hardin

Professor Edward B. Saff

Professor Gieri Simonett

Professor James H. Dickerson

Copyright c© 2010 by Justin Fitzpatrick

All Rights Reserved

This work is dedicated to my two mothers: to Mrs. Cynthia Fitzpatrick, who has

suffered more physically and emotionally during my five years of graduate school than

any woman ought to suffer in a lifetime; and to the Blessed Virgin Mary, whose help

I constantly sought and whose help is truly perpetual.

iii

acknowledgements

First and foremost I wish to thank my adviser, Professor Doug Hardin, for his tireless

efforts to help me throughout my graduate school career. He is a true gentleman

from whom I have learned a lot about mathematics and a lot about how to be a

mathematician. I believe he got the most out of me, and I cannot pay him any

higher compliment than that. It is my sincere hope that we will work together on

mathematics long into the future.

I wish to thank also the other members of my committee: Professor Ed Saff is a

giant in the field of applied analysis. It has been an honor and privilege to call myself

a member of a research group of which he is a part. He has been helpful to me in

my research many times, and he is always a pleasure to be around, not only for his

brilliance but also for his camaraderie. Professor Gieri Simonett taught me analysis

in my first year of graduate school and he is a gifted instructor as well as a dedicated

one. His professionalism and high standards are standards that I try to meet with my

own work. Professor Emmanuele DiBenedetto is a man I admire. He imparted words

of wisdom about mathematics and about life as a mathematician I can never forget.

I cannot say enough about his patience in the classroom and his dedication outside

of the classroom. I wish to thank also Professor Jay Dickerson of physics. He kindly

agreed to serve as a member of my committee and during my qualifying defense gave

me poignant questions to consider for future research.

The graduate students at Vanderbilt University truly form a community and I

could list almost every single graduate student I have met during my career as a

person to whom I am in some form of debt. In the interest of brevity let me mention

just a few: Dr. Matt Calef and Dr. Abey Lopez are two of the greatest friends I made

in my life. They are the gold standards of graduate students, and I always sought to

follow them even to the point that I joined their field of research. Ms. Tara Davis

iv

and Mr. Jeremy LeCrone have also been great friends to me and have always been

there for me when I have been in need. I want to thank Mr. Wes Camp for helping

me generate the figure in Chapter 3. I regret that I do not have enough space here to

thank personally the many other graduate students who hold a special place in my

heart.

Finally I wish to thank my family, without whom I could have accomplished

nothing. My father, Mr. Sean Fitzpatrick, and my mother, Mrs. Cynthia Fitzpatrick,

worked tirelessly every single day since before I was born to put me in a position to

succeed in life. I love them very much and I am only sorry that I have not done more

in my life to make them proud. My brother, Dr. Neal Fitzpatrick, always brought

me a smile when life was difficult for me and helped me to stay calm when I let my

emotions get out of control. I thank him for his brotherhood and his friendship these

26 years.

v

table of contents

Page

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter

I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

II.1 The Riesz Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8II.2 The Continuous Energy Minimization Problem . . . . . . . . . . . 10II.3 Linking the Discrete and Continuous Problems . . . . . . . . . . . 14II.4 The Case s > d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18II.5 Lattice Configurations . . . . . . . . . . . . . . . . . . . . . . . . . 22II.6 Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 25II.7 Bonnesen Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 31

III NEW GEOMETRIC INEQUALITIES AND A LOWER BOUND FOR CS,2 34

III.1 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 34III.2 The Constant Cs,d . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

IV GEOMETRIC CONSTRAINTS ON CERTAIN CONFIGURATIONS . . 52

IV.1 Some Results Concerning the Geometry of s-Optimal Configurationsfor Large s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

IV.2 Analogous Results for S2 . . . . . . . . . . . . . . . . . . . . . . . 58IV.3 Results in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 64IV.4 Holder Inequality Techniques and Refined Estimates . . . . . . . . 69

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vi

list of figures

Figure Page

1 A portion of a polygon and its quantities of interest . . . . . . . . . . . . 34

vii

chapter i

introduction

The problem of placing a fixed number of electrons on the 2-sphere in such a way so

that the total electrostatic energy is minimized has been of interest since at least 1904

and Thomson’s article ([36]). This problem and subsequently posed associated prob-

lems are problems with wide-ranging applications including approximation theory,

best-packing, viral morphology, and coding theory, to name just a few.

Given two electrons placed at x1 and x2 in R3, the electrostatic potential energy

between them is, up to a constant,

1

|x1 − x2|.

Furthermore, the total energy required to assemble a collection of N electrons at x1,

x2, . . . , xN in R3 is, up to the same constant,

N∑i=1

N∑j>i

1

|xi − xj|. (1)

If the points x1, x2, . . . , xN are now constrained to lie in some set A, then the

question of what configuration of those points minimizes the quantity in (1) is called

a minimal energy problem for the set A; this is precisely Thomson’s problem when

A = S2 ⊆ R3. Over a century after his posing of the problem, solutions to Thomson’s

problem are known only for N = 2, 3, 4, 6, 12, and more recently N = 5 (see ([38]),

([22]), ([1]), ([33])).

The difficulty of the minimal energy problem even for relatively small values of N

has led to several different approaches that attempt to circumvent the computational

1

difficulties. The primary approach considered here is an asymptotic one: Given data

about the growth of the total electrostatic energy as the number of electrons grows

to infinity, what can be said about the minimal energy configurations themselves?

More generally, one can consider the problem of minimizing over all N -point

configurations on a set A the quantity

N∑i=1

N∑j>i

k(xi, xj),

where k is a lower semicontinuous function on A called a kernel. Of particular

interest in this paper is the kernel

k(x, y; s) := ks(x, y) :=1

|x− y|s, (2)

which is called the Riesz kernel of exponent s for s > 0. Notice that taking the

s-th root of the Riesz s-energy

N∑i=1

N∑j=1

j 6=i

1

|xi − xj|s

which is the energy associated to the Riesz kernel of exponent s, and then taking the

limit as s goes to infinity leaves only the largest term in (2). In this sense the minimal

energy problem approximates the best-packing problem as s approaches infinity.

Fejes-Toth showed in 1940 in ([17]) that the best-packing problem in the plane

is solved by the hexagonal lattice; that is, the densest way to pack non-overlapping

disks in the plane is to place them so that their centers form a hexagonal lattice.

The density of a circle-packing indicates the fraction of space covered by the disks

used in the packing; Fejes-Toth’s result implies that the maximum packing density is

π√12

. Given the solution in dimension two to the best packing problem and the above

2

remark about how the best-packing problem is, in that sense, approximated by the

minimal energy problem, we seek to gain information about minimal Riesz s-energy

configurations in subsets of the plane for large values of s; more precisely, that they

share many characteristics with the hexagonal lattice itself.

The following result of Gruber was motivational in our efforts to generate results

about the geometry of minimal and near-minimal Riesz energy configurations. In

particular, all of the results of Chapter 4 are in the same vein.

Theorem 10. ([19]) Let S ⊆ R2 be a finite point set and let δ > 0. A point p ∈ S is

said to be the center of a regular hexagon in S up to δ if there are σ > 0, the

edge length of the hexagon, and points p1, . . . , p6 in S so that

S ∩ {x : |x− p| ≤ 1.1σ} = {p, p1, . . . , p6}

|p− pi|, |pi+1 − pi| ≤ (1± δ)σ, i = 1, . . . , 6,

where (1 ± δ)σ means a quantity between (1 − δ)σ and (1 + δ)σ. Now let H be a

convex 3-, 4-, 5-, or 6-gon and ε > 0 sufficiently small. Then for all packings of H

by m congruent disks of sufficiently small radius and density more than π√12

(1 − ε),

the following hold: in the set of centers of these circles, each center, with a set of less

than 50ε1/3m exceptions, is the center of a regular hexagon up to 500ε1/3. All these

hexagons have the same edge length (in the above sense).

To study the geometry of point configurations in the plane, the Voronoi diagram

associated to a point configuration is introduced. Stated simply, the Voronoi diagram

is a collection of Voronoi cells, and a Voronoi cell associated to a point x in the

configuration is the set of all points that are at least as close to x than to any other

3

point in the configuration. In the hexagonal lattice, for example, every Voronoi cell

is a regular hexagon.

What now follows are the main results of this thesis. The results as stated here will

be for minimal Riesz s-energy configurations ωsN on the unit square [0, 1]× [0, 1] ⊆ R2;

they are generalized somewhat later in the paper.

Theorem IV.1.1. Let δ(N) = 2√N√

12, C > 6 and γ ∈ (0, 1). Then there is s0 and

N0 = N0(s) such that for all N > N0,

|AsN,γ|N

≤ Cγs

for all s > s0, where

AsN,γ = {x ∈ ωsN : r(x) < γδ(N)}

and r(x) = min {|x− y| : x 6= y ∈ ωsN}.

Theorem IV.1.2. Let δ(N) = 2√N√

12. Fix γ ∈ (0, 1) and C > 6. Let BN,γ denote

the set of points whose Voronoi cell has an inradius of at least γδ(N)2

but does not

have exactly six sides. Then there is a positive constant D such that, for N = N(s)

and s sufficiently large,

|BN,γ|N

≤ D ·(

1

γ2− 1 + Cγs

).

Theorem IV.1.3. Let γ ∈ (0, 1), Γ > 1 be fixed, and let DN,γ,Γ denote those points

not in AsN,γ whose Voronoi cells are 6-sided and have area more than Γ/N . Then for

any C > 6, there are N = N(s) and s sufficiently large such that

|DN,γ,Γ|N

≤ 1− γ2 (1− Cγs)Γ− γ2

.

4

Given the Voronoi diagram associated to a minimal Riesz s-energy configuration

for s sufficiently large, these results indicate that the fraction of Voronoi cells that

are hexagonal with at least a certain inradius and at most a certain area is close to

1; in fact, as close to 1 as desired for correct choices of the parameters.

Later in Chapter 4, we go on to give the following improvement of Theorem IV.1.2:

Theorem IV.4.2. Let C > 6. Let B6 ⊆ ωsN denote the set of points in the unit

square whose Voronoi cell does not have six sides and does not meet the boundary of

the unit square. Then for s sufficiently large, there is N0 = N0(s) and a constant D

such that for all N > N0,

|B6|N≤ D

(C

2s+2 − 1

).

Notice that there is no dependence here on the parameter γ.

Denote by Es(A,N) the minimal N -point Riesz s-energy on the set A and by

Hd(·) d-dimensional Hausdorff measure normalized so that the measure of the unit

cube in Rd is 1. A result due to Hardin and Saff in ([20]) is that, for a large class of

d-dimensional sets A (to be defined later) and for s > d,

limN→∞

Es(A,N)

N1+ sd

=Cs,d

(Hd(A))sd,

where Cs,d is a constant that depends on s and d but not on the set A. This constant

is of interest because it is related to the geometry of these minimal s-energy config-

urations; more will be discussed on this matter later. It was shown in ([3]) that, for

all d > 1, lims→∞C1/ss,d exists and is given by

lims→∞

C1/ss,d =

1

2

(∆d

βd

) 1d

,

5

where ∆d is the greatest sphere packing density in Rd, and βd is the measure of the

unit ball in Rd. By removing the need to take the s-th root, in Chapter 3 we will

improve upon this result in the case d = 2:

Theorem III.2.4.

lims→∞

Cs,2(√3

2

) s2

= 6. (3)

One can understand here the 6 as reflecting the fact that most cells are 6-sided, and

the(√

32

)s/2as corresponding to the area of a fundamental cell of the hexagonal lattice

raised to the appropriate power s/2.

Establishing Theorem III.2.4 has been a goal of the author for some time, and

crucial in its proof was the establishment of the two geometric theorems of Chapter

3. Given a convex M -gon P and an interior point p, let {ri}Mi=1 be the lengths of its

altitudes drawn from p and A(P ) be its area. Let a(n) be the area of the regular

n-gon of inradius 1. We consider the quantity

F (P ) :=

(M∑i=1

r−si

)A(P )

s2

Theorem III.1.4. Let M ∈ N be given and P be a convex M -gon. Then there is

an s0 = s0(M) so that for all s > s0, F (P ) ≥ Ma(M)s2 , which is the value of F (P )

when P is the regular M -gon and p is the center.

Theorem III.1.6. For any convex M -gon P with interior point p and any s ≥ 2, we

have

F (P ) ≥ minν≤M

νa(ν)s2 .

Furthermore, this inequality is strict unless F (P ) = Ma(M)s2 .

6

For an M -gon P , now let ∆s(P ) := F (P ) −Ma(M)s2 , so that ∆s gives a measure

of difference between the polygon P and the regular M -gon. Then in Chapter 4 we

show the following:

Theorem IV.4.4. Let {ωsN} be a sequence of s-optimal configurations on the unit

square for s > 2. Then

lim supN→∞

1

N

N ′∑i=1

∆s(Vi)2s+2

≤ (ζΛ(s)− 6)2s+2 · a(6)

ss+2

where N ′ denotes a natural number less than N but that satisfies

limN→∞

N ′

N= 1.

Further, Λ denotes the hexagonal lattice, and ζΛ the associated Epstein zeta function:

ζΛ(s) =∑x∈Λ

x 6=0

1

|x|s.

These results indicate the significance of the regular figures in the minimal energy

problem, as will be discussed at length in what follows.

The organization of the remainder of this thesis is as follows: In Chapter 2, some

background information will be given about minimal energy problems and theorems

in plane geometry that are useful in gaining information about minimal energy con-

figurations on two-dimensional sets. In Chapter 3, we present two geometric theorems

that give insight into the local structure of minimal energy configurations. In Chapter

4, we demonstrate some new results detailing constraints on the geometry of optimal

and near-optimal configurations.

7

chapter ii

background

II.1 The Riesz Kernel

Let A ⊆ Rn. For s > 0, the Riesz kernel of exponent s, ks(x, y) : A× A→ R, is

defined to be

ks(x, y) =1

|x− y|s.

(We define it to be ∞ in the case x = y.) Sometimes for simplicity any such func-

tion ks(x, y), s > 0 is simply referred to as the Riesz kernel. For an infinite

compact set A ⊆ Rn, define the Riesz s-energy of a configuration of N points

ωN = {x1, x2, . . . , xN} ⊆ A to be

Es(ωN) =N∑i=1

N∑j 6=i=1

1

|xi − xj|s. (4)

(Notice here that the definition of energy differs by a factor of two from the definition

given in the introduction.) For a fixed N , the minimal energy problem on A associ-

ated to this kernel is the problem of finding an N -point configuration ωsN ⊆ A that

minimizes Es(ωN) as defined above over all N -point configurations ωN ⊆ A. Recall

that when s = 1, n = 3, and A = S2, this is simply Thomson’s problem.

What will be shown first is that, for any infinite compact set A, there is indeed a

configuration that minimizes the energy defined in (4).

8

Proposition II.1.1. Let A ⊆ Rn be an infinite compact set, s > 0 be given, and

N be any natural number. Then there is a configuration ωsN ⊆ A that achieves the

minimum for the quantity {Es(ωN) : ωN ⊆ A, |ωN | = N}.

Proof. Let ωN be any configuration of N distinct points on A. Since Es(ωN) is finite,

choose ε > 0 such that 1/εs > Es(ωN). For this value of ε, now set

kεs(x, y) =

1

|x−y|s , |x− y| ≥ ε

ε−s, |x− y| < ε.

(5)

Since Es(ωN) < 1/εs, we have by the definition of the Riesz kernel that for any

configuration ω′N of energy not more than Es(ωN), all of the points in ω′N are separated

by at least ε. Hence, denoting by Eεs the energy associated to the truncated Riesz

kernel defined in (5), we have that

Eεs(ω

′N) = Es(ω

′N) (6)

for all configurations ω′N of energy not more than Es(ωN). Notice now that the kernel

kεs is continuous, and so by the compactness of A, Eεs attains a minimum for some

N -point configuration, which shall be called ωsN . (6) now gives the theorem.

(The previous result holds in more generality, with the same proof if the energy is

defined with any lower semicontinuous kernel. In the proof the lower semicontinuity

of the Riesz kernel was used explicitly.)

For a given value of s, configurations ωsN that give this minimum will be called s-

optimal configurations. Notice that s-optimal configurations need not be unique;

9

for example, when A = S1, if ωsN is any s-optimal configuration, then any rotation of

ωsN is also s-optimal. With the result of Proposition II.1.1, now define

Es(A,N) = min {Es(ωN) : ωN ⊆ A, |ωN | = N}.

Given s > 0 and A, the questions of finding s-optimal configurations ωsN on A and

Es(A,N) quickly become intractable even for modest values of N . This is due to the

facts that the system of equations defining this minimum is intractable even when N

is small; there is strong evidence that there are many local minima for the energy that

are not global minima, and according to this evidence the number of local minima

increases exponentially with N . (To view some of this evidence see e.g. ([5]).)

An approach that has generated recent interest is to consider the asymptotics of

the energy Es(A,N) and the asymptotic distribution of s-optimal configurations ωsN .

The asymptotic distribution of such configurations depends highly on the value of s,

as we will show in the following sections.

II.2 The Continuous Energy Minimization Problem

Following ([12]), a Borel measure µ on Rn is said to be a Radon measure if µ(K)

is finite for every compact set K ⊆ Rn. An equivalent definition of a Radon measure

(see [15]) is that the Borel measure must be inner and outer regular; that is, for every

measurable set A,

µ(A) = sup {µ(K) : K ⊆ A is compact} = inf {µ(O) : O ⊇ A is open}.

Let A ⊆ Rn be a compact set of positive d-dimensional Hausdorff measure, and let

10

µ be a Radon probability measure on A; that is, a positive Radon measure satisfying

µ(A) = 1. For s > 0, we define the Riesz s-energy Is of a probability measure µ

supported on A:

Is(µ) :=∫A

∫A

1

|x− y|sdµ(x)dµ(y). (7)

Similar to the question of minimizing the discrete Riesz s-energy posed in the previous

section is the question of minimizing the continuous Riesz s-energy; that is, the

question of finding a probability measure µ on a given compact set A that minimizes

the quantity in (7) over all probability measures on A. Intuitively this is thought

of as finding the continuous distribution of a fixed amount of charge that minimizes

the total potential energy on a conductor rather than finding the locations of point

charges that minimize this energy.

As before, it requires verification that the problem of finding an energy-minimizing

measure does have a solution. For s ∈ (0, d), it is a consequence of Frostman’s Lemma

and the fact that the Riesz kernel is positive definite that there is a measure that

gives a finite minimum for (7). For more details, the reader is referred to ([24]).

Therein it is further shown that such a minimizing measure is unique. In this case,

the minimizing measure µs,A is referred to as the equilibrium measure on the set

A and its energy Is(µs,A) is called the Wiener energy of A. The quantity

Cs(A) =1

Is(µs,A)

is called the s-capacity of A and has been the subject of considerable study. Loosely

speaking, a unit charge will be able to be distributed more sparsely over a larger set,

and so such a set will generally have low Wiener energy and high s-capacity. For

more details the reader is again referred to ([24]) and also ([35]). Remarks on the

case s ≥ d will be made in a subsequent section.

11

For s ∈ (0, d), define now the s-potential of a probability measure µ at a point

x ∈ A to be

Uµs (x) =

∫A

1

|x− y|sdµ(y).

It will also be said that a property holds (s)-approximately everywhere on A if

the set of points where the property does not hold contains no compact subsets of

positive s-capacity.

A result that merits demonstration at this time is one about the potential of

the equilibrium measure. As its definition suggests, the equilibrium measure can be

thought of as a static distribution of charge; that is, the charge distribution that

admits no difference in electrostatic potential from one location to another, so that

potential has gradient zero, implying that the location of the charge remains fixed.

This intuition is confirmed by the following:

Proposition II.2.1. ([18]) Let A ⊆ Rn be compact, and Hd(A) > 0. For s ∈ (0, d),

let µs,A be the equilibrium measure on A. Then

• Uµs,As ≥ Is(µs,A) approximately everywhere on A,

• Uµs,As ≤ Is(µs,A) everywhere on the support of µs,A, and

• Uµs,As = Is(µs,A) µs,A-almost everywhere.

Before proving this proposition, we introduce a Hilbert space structure on the

space of probability measures on A. Given two probability measures µ and ν on A,

define the following bilinear form:

〈µ, ν〉 =∫A

∫A

1

|x− y|sdµ(x)dν(y).

12

This bilinear form defines an inner product; for more details, see ([24]). This inner

product induces a norm on the space of probability measures in the usual way:

||µ||2 = 〈µ, µ〉 =∫A

∫A

1

|x− y|sdµ(x)dµ(y).

It is shown in ([24]) that the space of probability measures is topologically complete

with respect to this norm. This is what will be necessary to prove Proposition II.2.1.

Proof of Proposition II.2.1. By Frostman’s Lemma, Is(µs,A) is finite. Let ν be an-

other probability measure with finite energy, and let a ∈ [0, 1]. Then ρa = aµs,A +

(1−a)ν is a probability measure, and since µs,A is the equilibrium measure, it follows

that

||µs,A||2 ≤ ||ρa||2 = a2||µs,A||2 + 2a(1− a)〈µs,A, ν〉+ (1− a)2||ν||2,

and so

||µs,A||2 = lima↑1

(1− a2)||µs,A||2 − (1− a)2||ν||2

2a− 2a2≤ 〈µs,A, ν〉. (8)

As ν(A) = 1, ||µs,A||2 · ν(A) ≤ 〈µs,A, ν〉 for all probability measures ν. Let now

N = {x ∈ A : Uµs,As (x) < ||µs,A||2}. Suppose that for some probability measure ν of

finite energy that ν(N) > 0. Then since ν is a Radon measure, there is a compact

set K ⊆ N with ν(K) > 0. Thus

∫KUµs,As dν < ||µs,A||2 · ν(K),

which together with (8) implies that ν has infinite energy. This proves the first item

in Proposition II.2.1. Letting ν = µs,A in the above argument shows in particular

that

13

µs,A([Uµs,As < ||µs,A||2]) = 0.

From here it follows that

||µs,A||2 =∫

[Uµs,As =||µs,A||2]

Uµs,As dµs,A +

∫[Uµs,As >||µs,A||2]

Uµs,As dµs,A

whence µs,A([Uµs,As > ||µs,A||2]) = 0. Since [U

µs,As > ||µs,A||2] is open, it is disjoint

from the support of µs,A, proving the second item of Proposition II.2.1. The third

item follows immediately from the first two.

II.3 Linking the Discrete and Continuous Problems

One can view the discrete problem as a special case of the continuous problem by

considering the measure placing a mass of 1N

at each point in an N -point configuration

ωN . The question of relating solutions to the continuous and discrete minimum energy

problems is a natural one and is the aim of this section.

Let A ⊆ Rp be compact. For a point x ∈ A, define the probability measure

δx(A) =

1, x ∈ A

0, x /∈ A;

and for a configuration of points ωN ⊆ A, define the probability measure

δωN =1

N

∑x∈ωN

δx.

The following result and proof can be found in e.g. ([24]):

14

Proposition II.3.1. Let A ⊆ Rp be a set of positive Hd-measure and let s ∈ (0, d).

Let {ωsN} be a sequence of s-optimal N -point configurations on A for N ≥ 2. Then

δωsN∗→ µs,A,

and

Es(A,N)

N2→ Is(µs,A)

as N →∞.

Proof. Let ωsN = {xi,N}Ni=1. We will show that the quantity

2

N(N − 1)Es(ω

sN) (9)

is increasing with N . Notice that

∑i<j

1

|xi,N − xj,N |s=

1

N − 2

N∑k=1

∑i<j

i 6=k,j 6=k

1

|xi,N − xj,N |s.

For each k, the inner sum above is the energy associated to an (N − 1)-point config-

uration on A. Therefore, the minimality of Es(ωsN−1) gives

∑i<j

1

|xi,N − xj,N |s≥ N

N − 2

∑i<j

1

|xi,N−1 − xj,N−1|s.

Multiplying on both sides of the above inequality by 2N(N−1)

gives

2

N(N − 1)

∑i<j

1

|xi,N − xj,N |s≥ 2

(N − 1)(N − 2)

∑i<j

1

|xi,N−1 − xj,N−1|s,

15

which establishes that the quantity in (9) is increasing with N . Therefore the limit

limN→∞

Es(A,N)

N2

exists as an extended real number.

Now let {yi}Ni=1 be an arbitrary configuration of N points on A. Then again by

the minimality of Es(ωsN) we have

∑i<j

1

|xi,N − xj,N |s≤∑i<j

1

|yi − yj|s.

Let µ be a probability measure onA. Integrating both sides with respect to dµ(y1) . . . dµ(yn)

gives

∑i<j

1

|xi,N − xj,N |s≤∑i<j

∫A

∫A

1

|yi − yj|sdµ(yi)dµ(yj),

since only the terms containing yi or yj are affected by integration with respect to

dµ(yi) and dµ(yj). Notice that each double integral in the sum above is the same,

whence

∑i<j

1

|xi,N − xj,N |s≤ N(N − 1)

2

∫A

∫A

1

|ya − yb|sdµ(ya)dµ(yb).

In particular, choosing µ to be the equilibrium measure µs,A and letting N tend to

infinity gives

limN→∞

2

N(N − 1)

∑i<j

1

|xi,N − xj,N |s≤∫A

∫A

1

|ya − yb|sdµs,A(ya)dµs,A(yb). (10)

Now fix ε > 0 and consider

16

∫A

∫Akεs(x, y)dδωsN (x)dδωsN (y).

Notice that, for ε sufficiently small, nonzero terms arising from distinct points in the

integral above are bounded above by ε−s, and terms arising from the same points give

a value of ε−s. Therefore, we have

∫A

∫Akεs(x, y)dδωsN (x)dδωsN (y) =

1

N2

∑i<j

1

|xi,N − xj,N |s+ε−s

N

≤ 2

N(N − 1)

∑i<j

1

|xi,N − xj,N |s+ε−s

N.

It is known (see e.g. ([24])) that the space of probability measures forms a weak-star

compact subspace of the space of all Radon measures on A. Therefore, we can choose

a weak-star cluster point ν of {δωsN}, and a subsequence of the natural numbers {Ni}

so that δωsNi∗→ ν as i→∞. Letting Ni tend to infinity and using (10) gives

∫A

∫Akεs(x, y)dν(x)dν(y) ≤

∫A

∫A

1

|ya − yb|sdµs,A(ya)dµs,A(yb).

Now let ε decrease to 0. Since the Riesz kernel is integrable for s < d, we have

Is(ν) ≤ Is(µs,A),

and the uniqueness of the equilibrium measure now gives the result.

The previous proposition holds in greater generality for other kernels that have

a unique equilibrium measure, optimal discrete configurations, and can be approxi-

mated by continuous functions; for more details see ([7]), ([14]), and ([18]).

17

II.4 The Case s > d

In the previous section it was seen how the continuous minimal energy problem can

provide insight into the asymptotics of the discrete minimal energy problem. It was

critical for this analysis that the exponent s of the Riesz kernel was less than the

Hausdorff dimension d of the set on which the minimal energy problem is being

considered; indeed, when s ≥ d, the continuous minimal energy problem does not

make sense in that the s-energy of any probability measure on a set of Hausdorff

dimension d is infinite. What follows is a proof of this fact for sets with finite d-

dimensional Hausdorff measure.

Theorem II.4.1 (Theorem 8.7 in ([26])). If Hs(A) < ∞, then Is(µ) = ∞ for all

probability measures µ supported on A.

Proof. The proof is by contradiction. Suppose that for a probability measure µ sup-

ported on a set A of Hausdorff dimension d, Hs(A) < ∞ and Is(µ) < ∞. (Recall

that, by definition of the Hausdorff measure Hs, when s > d, Hs(A) = 0.) Since

Is(µ) =∫ ∫ 1

|x− y|sdµ(x)dµ(y) <∞,

Tonelli’s theorem implies that the inner integral is finite for µ−a.a. x in A. It follows

from the Lebesgue Differentiation Theorem (see e.g. p. 38 of [26]) that, for these x,

limr↓0

∫B(x,r)

1

|x− y|sdµ(y) = 0.

Now apply Egorov’s Theorem to select A0 ⊆ A such that µ(A0) > 1/2 and the above

limit is uniform on the set A0. Fix ε > 0. Then by the uniformity of the above limit

we may find r0 such that, for all x ∈ A0 and r < r0, it follows that

18

µ(B(x, r)) · r−s ≤∫B(x,r)

1

|x− y|sdµ(y) < ε,

whence µ(B(x, r)) < εrs. Using the construction of the Hausdorff measure, there is

a collection of sets {Ui}∞i=1 satisfying:

(a) A0 ⊆∞⋃i=1

Ui,

(b) A0 ∩ Ui 6= ∅ for all i,

(c) diam Ui < r0,

(d)∞∑i=1

(diam Ui)s < Hs(A0) + 1.

Now select from each Ui a point xi lying also in A0, and let ri = diam Ui. Then one

has

1

2< µ(A0) ≤ ε

∞∑i=1

rsi < ε (Hs(A0) + 1) .

As ε is arbitrary, it follows that Hs(A0) is infinite, contradicting the assumption.

Since it follows from this result (and Proposition II.3.1) that

limN→∞

Es(A,N)

N2=∞,

19

it follows that the order of growth of the N -point minimal energy is larger than N2

when s is larger than the Hausdorff dimension of A. Intuitively one can think of this

as the Riesz kernel having an increasingly local character as s increases (a remark

along these lines is made in the introduction). Alternatively, one can also view this

as an extension of the non-integrability of

∫B(0,1)

1

|x|sdx,

where dx denotes Lebesgue measure in Rd. For this reason the case s ≥ d is often

referred to as the hypersingular case, while the case s < d is referred to as the potential

theory case.

Results for the asymptotics of the energy in the case s > d came later. These

results are for a class of sets that are defined below. For more information on sets of

this type, the reader is referred to ([13]).

Definition II.4.2. A set A is said to be a d-rectifiable manifold if it can be written

A =n⋃i=1

φi(Ki)

where Ki is a d-dimensional compact set and φi is a bi-Lipschitz function on an open

set Gi ⊇ Ki.

Definition II.4.3. A set A is said to be a d-rectifiable set if it is a Lipschitz image

of a bounded set in Rd.

In ([20]), Hardin and Saff were able to show the following:

Theorem II.4.4. ([20], Theorem 2.4) If A ⊆ Rp is a d-rectifiable manifold that is

also a compact subset of a d-dimensional C1-manifold, then

20

limN→∞

Ed(A,N)

N2 logN=Hd(Bd)Hd(A)

, (11)

where Bd is the closed unit ball in Rd. Furthermore, if Hd(A) > 0, then for any

sequence {ωdN} of configurations on A satisfying (11), we have

limN→∞

1

N

∑x∈ωdN

δx =Hd(·)|AHd(A)

,

where the convergence above is understood in the weak-star topology of measures;

that is, for every continuous function f on A,

limN→∞

1

N

∑x∈ωdN

f(x) =1

Hd(A)

∫Af(x)dHd(x).

Theorem II.4.5. ([20], Theorem 2.4) If A ⊆ Rp is a d-rectifiable manifold, then for

s > d,

limN→∞

Es(A,N)

N1+ sd

=Cs,d

(Hd(A))sd, (12)

where Cs,d is a constant depending on s and d but not on the set A. Furthermore,

for any sequence {ωsN} of configurations on A satisfying (12), we have

limN→∞

1

N

∑x∈ωsN

δx =Hd(·)|AHd(A)

, (13)

and the convergence is again understood in the weak-star topology of measures.

These results establish the order of growth of the energy for s ≥ d and further

establish that configurations that are asymptotically optimal (in the sense that they

satisfy (11) or (12)) are also asymptotically uniformly distributed on the set A, a

21

condition that is not generally satisfied in the potential theory case. The fact that

Riesz s-optimal configurations are asymptotically uniformly distributed in the hyper-

singular case will be used frequently in later chapters.

In ([3]), Borodachov, Hardin, and Saff were able to generalize Theorem II.4.5 to

d-rectifiable sets. Another result of ([20]) that is noteworthy is the following:

Theorem II.4.6. ([20], Theorem 2.3) Let A ⊆ Rd be an infinite compact set, and

let s ≥ d, N ≥ 2. Let ωsN an s-optimal configuration on A. Then there is a positive

constant C = C(A, s, d) so that

minxi 6=xj∈ωsN

{|xi − xj|} ≥

CN

−1d , s > d

C(N logN)−1d , s = d.

It should be noted that it is believed that the logN term above is not necessary in the

case s = d. These separation results will be compared to remarks about circle-packing

in section II.6.

Other cases where s is non-positive are also worth mentioning. Included among

these is the case s < 0, in which one considers a maximization problem. Results in

this case were obtained by Bjorck in ([2]). The case “s=0” is not the Riesz kernel

with exponent 0, but rather the Riesz kernel is replaced by the kernel k0(x, y) =

− log |x− y|. The techniques of potential theory apply in this case; a standard text

containing results is ([35]). Moreover, results for complex s were obtained recently

by Brauchart, Hardin, and Saff; see ([4]) for more details.

II.5 Lattice Configurations

The increasingly local character of the Riesz kernel of exponent s together with the

local homogeneity of lattice configurations makes lattice configurations natural candi-

22

dates to consider when searching for s-optimal configurations. Here an n-dimensional

lattice L in Rn will be defined to be

L =

{n∑i=1

civi : ci ∈ Z}, (14)

where {vi} is a collection of n linearly independent unit vectors in Rn, called the

basis vectors of the lattice. Intersecting such a lattice with a compact set A on

which a minimal energy problem is to be considered is natural, and scaling the lattice

allows one to consider configurations with an increasing number of points and hence

asymptotics. This matter will be discussed in greater detail in Chapter 3.

The question of what lattice configuration of points minimizes a certain energy

is an important one and has been studied extensively. Important in such studies are

the Epstein zeta function associated to a lattice:

ζL(s) =∑x∈Lx 6=0

1

|x|s

and the theta function associated to a lattice:

θL(s) =∑x∈Lx6=0

e−2πs|x|.

Cassels ([6]) and Rankin ([31]) gave partial results about minimizing the Epstein

zeta function over all two-dimensional lattices subject to the constraint that the

fundamental cell have area 1. Here the fundamental cell of a lattice L is defined to

be the closed convex hull of

{n∑i=1

civi : ci = 0 or 1

},

23

where again vi denotes a basis vector of the lattice.

Montgomery observed in ([27]) that if ξL(s) = ζL(s)Γ(s)(2π)−s, then

ξL(s) =1

s− 1+

1

s+∫ ∞

1(θL(α)− 1)(αs + α1−s)dα,

which provides a link between solving the problem of finding the minimal lattice for

theta functions and for zeta functions. Using this approach, in ([27]) he was able

to show that the hexagonal lattice in R2, which has basis vectors (1, 0) and (12,√

32

),

minimizes the theta function and the zeta function over all lattices subject to the

same constraint on the fundamental cell for all real s > 0.

In ([9]), Cohn and Kumar generalized the problem slightly and sought to find

periodic point configurations (unions of finitely many translates of a lattice) that are

“universally optimal,” where they define universally optimal in the following way:

Suppose we are given a function

f := f(|x− y|2) : (0,∞)→ R,

called the potential function, which is completely monotone; that is, the function

is positive, decreasing, and its subsequent derivatives alternate signs (and are all

defined). Then a periodic point configuration P is called universally optimal if it

minimizes

∑x,y∈Px 6=y

f(|x− y|2)

for all completely monotone functions f . Numerical evidence found in ([8]), and their

own work in ([9]) cause Cohn and Kumar to conjecture that the hexagonal lattice

is universally optimal in R2, that the E8 root lattice is universally optimal in R8,

24

and that the Leech lattice is universally optimal R24. (For more details on these

second two lattices, the reader is referred to ([8]) and ([10]), respectively.) Sarnak

and Strombergsson were able to show in ([34]) that these lattices are local minima for

energy (as defined by Cohn and Kumar) amongst lattice configurations, and recent

work by Coulangeon and Schurmann in ([11]) shows that these lattices are indeed local

minima for this energy for more general perturbations of the lattice. The numerical

evidence in ([8]) seems to indicate that in dimensions other than 2, 8, and 24, there

are no universally optimal configurations; Cohn and Kumar remark in ([9]) that there

are certainly no universally optimal configurations in dimensions 3, 5, 6, and 7.

II.6 Voronoi Diagrams

As mentioned earlier, because the discrete minimum energy problem is so difficult

to solve, relatively little is known about the geometric structure of s-optimal con-

figurations. This section introduces the Voronoi diagram associated to an N -point

configuration, after which a few facts that shall be used in the following chapters

are demonstrated in order to give some restrictions on the Voronoi diagram of an

s-optimal N -point configuration.

First formally define the Voronoi cells and Voronoi diagram associated to any

N -point configuration in R2:

Definition II.6.1. Let ωN = {x1, x2, · · · , xn} be a collection of N points in R2.

Then for any i = 1, 2, · · · , n, the Voronoi cell associated to xi is given by

V (xi) =⋂

1≤j 6=i≤n{x ∈ R2 : |x− xi| ≤ |x− xj|} (15)

The points xi are called the centers of their respective Voronoi cells. Each set

in the intersection in (15) is the half-space (containing xi) whose boundary is the

25

perpendicular bisector of the segment connecting xi to xj. Since there are finitely

many points in the configuration, and each Voronoi cell is an intersection of half-

spaces, one readily sees that V (xi) is a convex polygon containing xi. If two Voronoi

cells share an edge, then the centers of those cells are said to be nearest neighbors.

Definition II.6.2. With ωN and {V (xi) : 1 ≤ i ≤ n} defined as in the previous

definition, the Voronoi diagram associated to ωN is given by

V(ωN) = {V (x1), V (x2), · · · , V (xn)}.

For a compact subset A ⊆ R2, the Voronoi diagram on A associated to ωN ⊆ A is

simply

V(ωN) ∩ A.

It is a standard fact that the Voronoi diagram is the dual graph of the Delaunay

triangulation. For more details on Voronoi diagrams, the reader is referred to ([37]).

The following is a simplified version of a theorem found in the text of Fejes-Toth

([16]):

Theorem II.6.3. ([16]) Let H denote a convex polygon of no more than 6 sides.

Then, given r > 0, the number N of discs of radius r that can be placed without

overlapping into H satisfies the inequality

N ≤ Area(H)

Area(h),

where h is a regular hexagon circumscribed about the disc of radius r.

26

The proof of Theorem II.6.3 rests on a pair of lemmas, which will be stated shortly.

First, recall that a real-valued function f(x) defined on an interval I is said to be

convex if, for all t ∈ [0, 1] and all x, y ∈ I,

f((1− t)x+ ty) ≤ (1− t)f(x) + tf(y). (16)

Graphically, we view this as follows: Given any two points (x, f(x)), (y, f(y)) on the

graph of f , every point on the graph of f whose abscissa lies between x and y lies below

(or possibly on) the segment connecting (x, f(x)) to (y, f(y)). When the inequality

in (16) is strict, the function f is said to be strictly convex. A sufficient condition

for a function to be (strictly) convex on an interval is for the second derivative to be

non-negative (positive) on that interval.

A useful fact about convex functions of which we will make use is the following:

Jensen’s Inequality (Special Case). Suppose that f : I → R is a convex function

on an interval I. Then for any x1, x2, . . . , xn in I,

n∑i=1

1

nf(xi) ≥ f

(∑ni=1 xin

). (17)

The inradius of a polygon is the radius of the largest circle that can be inscribed

in that polygon. It is a simple exercise in geometry to show that the area of a regular

n-gon of inradius 1 is given by

a(n) = n · tan(π/n).

It is known (see [16]) that the n-gon of minimal area with inradius 1 is the regular

n-gon circumscribed about a circle of radius 1. This fact will be useful in what follows.

Lemma II.6.4. Let a(x) = x · tan(π/x). Then a(x) is strictly convex on [3,∞).

27

Proof. The proof is an exercise in basic calculus:

a′(x) = tan(π/x)− π sec2(π/x)

x

a′′(x) = −π sec2(π/x)

x2+ π

sec2(π/x)

x2+

(π2

x3

)(2sec2(π/x)tan(π/x))

=

(π2

x3

)(2sec2(π/x)tan(π/x)),

which is positive for x ≥ 3.

The next lemma gives an upper bound on the number of edges contained in a

Voronoi diagram on a polygon of six or fewer sides.

Lemma II.6.5. ([16]) Let V(ωN) = {V (x1), V (x2), · · · , V (xN)} be the Voronoi dia-

gram on a convex polygon H ⊆ R2 with at most six sides associated to ωN ⊆ H, and

for all 1 ≤ i ≤ N , let νi denote the number of sides of V (xi). Then

N∑i=1

νi ≤ 6N.

Proof. Denote by v and e the number of vertices and edges of the Voronoi diagram

on H. Ignoring the unbounded face of the planar graph determined by the Voronoi

diagram on H, apply Euler’s formula for planar graphs:

v − e+N = 1. (18)

Denote now by a the number of vertices on the boundary of H from which at least

three edges emanate and by b the number of vertices on the boundary of H from which

two edges emanate. Notice that 2e will count each vertex from which at least three

28

edges emanate at least three times and each vertex from which two edges emanate

twice. Hence,

3v ≤ 2e+ b. (19)

Notice further that, since every edge in the interior of H is shared by precisely two

Voronoi cells,∑Ni=1 νi will count each edge of the Voronoi diagram twice except for

those on the boundary. Since the number of vertices on the boundary equals the

number of edges on the boundary, we have:

N∑i=1

νi = 2e− (a+ b). (20)

Now (18), (19), and (20) will be combined. (18) gives

3v = 3e+ 3− 3N,

whence (19) implies

3e+ 3− 3N ≤ 2e+ b,

implying

2e ≤ 2b+ 6N − 6.

Combining this with (20) gives

N∑i=1

νi = 2e− a− b ≤ 6N − a+ b− 6.

29

Notice now that if a vertex has only two edges emanating from it, it must be a vertex

of the polygon H. It follows that b ≤ 6, from which it follows that

N∑i=1

νi ≤ 6N.

The previous two lemmas now give the result of Theorem II.6.3 rather quickly:

Proof of Theorem II.6.3. Suppose N discs are packed into the convex polygon H.

Let ωN = {xi}Ni=1 denote the centers of these discs. Form the Voronoi diagram

V(ωN) = {V (x1), V (x2), · · · , V (xN)} on H associated to the configuration ωN , and

suppose the Voronoi cell V (xi) has νi sides. By construction, each Voronoi cell will

have an inradius of at least r, whence

Area(H) =N∑i=1

Area(V (xi)) ≥ r2N∑i=1

a(νi).

Now apply the convexity result of Lemma II.6.4:

Area(H) ≥ r2N∑i=1

a(νi) ≥ r2Na

(∑Ni=1 νiN

)≥ r2Na(6), (21)

where the last inequality is a consequence of Lemma II.6.5. To conclude we simply

observe that the area of a regular hexagon circumscribed about a disc of radius r is

r2a(6).

Theorem II.6.3 has a natural application to best-packing. Let A denote the unit

square in R2. We wish to maximize the following quantity over all N -point configu-

rations ωN ⊆ A:

30

δ(ωN) = min{|x− y| : x 6= y ∈ ωN}

Call this maximum δ(N). Then solving (21) for r, one has that δ(N) ≤ 2√Na(6)

,

and taking N sufficiently large, one can make this inequality as close to an equality

as desired. (The factor 2 reflects the difference between the inradius of a Voronoi

cell and the distance between points whose Voronoi cells share an edge.) This indi-

cates that the packing radius of N points into the unit square is, for N large, close

to 2√Na(6)

. (Notice the concordance of this result with the separation estimates of

Theorem II.4.6.) This result will be seen again in Chapter 4.

II.7 Bonnesen Inequalities

The utility of considering the concept of Voronoi diagram presented in the previous

section as a means of investigating a minimal energy problem is quite simple: Since

the distances between certain points (nearest neighbors) in a configuration is double

the perpendicular distances from the center of a Voronoi cell to its respective edges,

the question of finding a configuration that minimizes energy on a planar set can

be studied (particularly for a kernel with predominantly local behavior) by finding a

convex polygon that is, in a certain sense, optimal. Geometric questions of this type

go back as far as the classical isoperimetric inequality: Given a simple closed planar

curve C with length L and enclosed area A,

L2 − 4πA ≥ 0,

with equality if and only if C is a circle. The following is known as the Bonnesen

Inequality (see e.g. ([29])):

31

L2 − 4πA ≥ π2(R− r)2,

whereR = inf {radii of circles containing C}, and r = sup {radii of circles contained in C}.

In general, a Bonnesen-style inequality is of the form

L2 − 4πA ≥ B,

where B ≥ 0 with equality only if C is a circle, and B has some geometric significance.

The following isoperimetric inequality for polygons is also well-known; see e.g.

([21]), ([30]): For an n-sided polygon Pn with perimeter Ln and area An,

L2n − 4n tan

(π

n

)An ≥ 0, (22)

with equality if and only if Pn is regular. Notice that the classical isoperimetric

inequality can be viewed as a limiting case of (22).

In 1997, Zhang in ([39]) proved some Bonnesen-style isoperimetric inequalities for

polygons using basic analytic inequalities. These inequalities are of the form

L2n − 4n tan

(π

n

)An ≥ Bn,

where Bn ≥ 0 with equality only if Pn is regular, and Bn has some geometric signifi-

cance. As an example, he demonstrates that

L2n − 4n tan

(π

n

)An ≥ (ln − Ln)2,

where ln is the perimeter of the regular n-gon with the same circumradius as Pn.

Equality holds only when Pn is regular.

32

Since the interest for one considering minimal energy problems concerns the

lengths of altitudes rather than perimeter, another result of Zhang in ([39]) that

merits consideration is the following inequality: Let {ri}ni=1 denote the n altitudes of

a polygon Pn, and set Rn =∑ni=1 ri. Then

R2n − n cot

(π

n

)≥ (Rn −Rn)2,

where Rn is the value of Rn for the regular n-gon having the same circumradius as

Pn. Equality holds only when Pn is regular.

Inequalities of this type could prove useful since they not only describe a polygon

that is, in various senses, optimal, but they also describe and quantify any devi-

ations from optimality. In the minimal energy setting, this could prove useful in

the description of configurations that are not only optimal configurations but merely

near-optimal.

In Chapter 3, certain inequalities will be established relating to the minimization

of the quantity

(n∑i=1

r−si

)(A(Pn))

s2

over all convex n-gons Pn (for fixed n). Zhang’s results do not immediately give any

bounds on this quantity; in fact, in Chapter 3 we will see that there are settings in

which the regular n-gon is not the minimizer for this quantity.

33

chapter iii

new geometric inequalities and a lower boundfor cs,2

III.1 Geometric Inequalities

For large values of s, information about s-optimal configurations’ geometry and energy

can be gotten by considering only nearest neighbors, due to the increasingly local

character of the Riesz s-energy kernel for these values of s. The natural question

then is the question of what nearest-neighbor structure is optimal in some appropriate

sense. This is the question considered in this section.

For a given M ∈ N, M ≥ 3, s ≥ 2, and a convex polygon P of M sides, let p be any

point interior to P , and label the vertices of P {v1, v2, . . . , vM} in a counterclockwise

manner. In what follows we shall identify a polygon P simply as the M -tuple of its

vertices (v1, . . . , vM). Then let hi be the segment from p perpendicular to the line

coinciding with the edge with endpoints vi and vi+1 (here vM+1 = v1). Let A(P ) be

the area of P .

vi vi + 1

p

xi xi + 1hi

µi-

µi+

Figure 1: A portion of a polygon and its quantities of interest

34

Consider the following (scale-invariant) quantity:

F (P ) := F (v1, . . . , vM) :=

(M∑i=1

r−si

)A(P )

s2 . (23)

where ri denotes the length of the segment hi. Notice that this quantity is also

translation-invariant, and so without loss of generality, we place p at the origin. As

implied at the beginning of this section, it is of interest to get a lower bound for F (P )

over all convex polygons P of M sides containing the origin.

Notice that the value of the quantity in (23) when P is the regular M -gon and p

is the center of P is

(M∑i=1

r−si

)A(P )

s2 = Ma(M)

s2 ,

where a(n) = n tan(πn

), which is the area of a regular n-gon with inradius 1. Prelim-

inary analysis initially seems to indicate that this may be the desired lower bound;

however, this is only partially the case, as will be discussed in Theorem 1.

Let now xi be the segment from p to vi. Denote by θ−i the angle formed by hi and

xi, and by θ+i the angle formed by hi and xi+1. Then we have that

A(P ) =1

2

2M∑i=1

s2i tan ti,

where s = (r1, r1, r2, r2, . . . , rM , rM) and t = (θ−1 , θ+1 , θ

−2 , θ

+2 , . . . , θ

−M , θ

+M). Notice that

it is possible that either θ−i or θ+i is negative, but both cannot be; indeed θ−i + θ+

i ∈

(0, π).

Notice further that tan θ−i + tan θ+i ≥ 2 tan

(θ−i +θ+

i

2

). Indeed, if both θ−i and θ+

i

are positive, this follows from the convexity of the tangent function on (0, π2). If one

of these two angles is negative, then one can consider the function

35

d(θ) = tan θ0 + tan θ − 2 tan

(θ0 + θ

2

),

where θ0 is negative and θ ∈ [−θ0,π2). Then d(−θ0) = 0 and

d′(θ) = sec2 θ − sec2

(θ0 + θ

2

)> 0,

since the secant function is increasing. We therefore have that

A(P ) =1

2

2M∑i=1

s2i tan ti ≥

M∑i=1

r2i tan θi,

where θi = 12(θ−i + θ+

i ) ∈ (0, π2). This allows us to bound the quantity in (23) from

below:

(M∑i=1

r−si

)A(P )

s2 ≥

(M∑i=1

r−si

)(M∑i=1

r2i tan θi

) s2

. (24)

This preliminary lower bound will be used in what follows. It will also be necessary

to consider degenerate cases in which adjacent vertices coincide, and so not all vertices

of the polygon will be required to be distinct. This motivates the following formulation

of the same minimization problem for polygons in this degenerate case:

If vi 6= vi+1, then hi will be defined as before. However, if vi = vi+1, we let hi

be the segment from p to vi, and ri be the length of hi. Then we simply view the

minimization of (23) as the minimization of a function F : R2M → R.

If ri = 0 then we will say that F (P ) = ∞. In addition, the scale-invariance of

the problem assures us that we can keep the vertices inside a compact subset of the

plane (for example, the unit disc). Therefore we will define the minimization problem

on the set of all M -tuples of vertices (v1, . . . , vM) satisfying the condition that the

vertices (v1, . . . , vM) form a counterclockwise-ordered list of (not necessarily distinct)

36

vertices of a convex M -gon contained in the unit disc and containing the origin. We

will call this set VM .

For collections of vertices V := (v1, . . . , vM) and W := (w1, . . . , wM), define a

metric D to be

D(V,W ) =n∑i=1

|vi − wi|,

where | · | denotes the usual Euclidean norm in R2. Then we have the following:

Lemma III.1.1. F (P ) is lower semicontinuous on VM . Hence it attains a minimum

in VM .

Proof. Fix P = (v1, . . . , vM) ∈ VM , and let Pn ∈ VM converge to P in the metric

defined above. If none of the vertices coincide, lower semicontinuity is a consequence

of the continuity of all the functions appearing in (24). If a pair of vertices vi and vi+1

do coincide, lower semicontinuity follows from the fact that ri is at least as large as

any perpendicular length drawn from p to any line through vi. That is, for a sequence

of polygons Pn ∈ VM , Pn = (v1,n, . . . , vM,n), the quantity

(M∑i=1

r−si

)A(P )

s2

is less than or equal to the value obtained by choosing ri to be the distance from p

to vi rather than the limiting value of distances from p to any line formed by distinct

points vi,n and vi+1,n. This establishes lower semicontinuity; the second statement is

a standard fact from real analysis.

For a natural number M ≥ 3,

37

fs(M) := inf

{(M∑i=1

r−si

)(A(P ))s/2

}= min

P∈VMF (P ), (25)

where the infimum is taken over all convex M -gons P and all interior points p, as

before.

Before proceeding, we note the following useful proposition and lemma:

Proposition III.1.2. The function gs(θ) := (tan θ)ss+2 is convex on (tan−1

(1√s+1

), π/2)

and concave on (0, tan−1(

1√s+1

)).

Proof. It suffices to check the sign of the second derivative of gs:

gs(θ) = tanss+2 θ

g′s(θ) =s

s+ 2tan

−2s+2 θ sec2 θ

g′′s (θ) =2s

s+ 2sec2 θ tan

−s−4s+2 θ

(tan2 θ − 1

s+ 2sec2 θ

)

=2s

(s+ 2)2sec2 θ tan

−s−4s+2 θ

((s+ 1) tan2 θ − 1

),

whence the result follows.

In the following, let γs := tan−1(

1√s+1

).

Lemma III.1.3. For M ∈ N, M ≥ 3, let h(θ, φ) : [0, γs]× [ πM, π

2)→ R be defined to

be

38

h(θ, φ) =tan

ss+2 θ + tan

ss+2 φ

2− tan

ss+2

(θ + φ

2

).

Then for s sufficiently large, h attains its minimum value at (θ1, θM) = (0, πM

), and

this minimum value is positive.

Proof. First fix θ. Taking the derivative of h with respect to φ gives

∂h

∂φ=

s

2(s+ 2)tan

−2s+2 φ sec2 φ− s

2(s+ 2)tan

−2s+2

(θ + φ

2

)sec2

(θ + φ

2

).

For large s, the tangent terms are approaching 1 since φ is bounded away from 0.

Since the secant function is increasing, and φ > θ+φ2

, this partial derivative is positive

for s sufficiently large, so for any fixed θ, h(θ, φ) has its minimum at φ = πM

.

Now minimize h(θ, π

M

)over θ:

∂h

∂θ=

s

2(s+ 2)tan

−2s+2 θ sec2 θ − s

2(s+ 2)tan

−2s+2

(θ + π

M

2

)sec2

(θ + π

M

2

).

This is infinite at θ = 0 and so the function h(θ, πM

) is increasing at first. What

we will show is that the derivative has a unique zero, which must be the location of

a local maximum for h(θ, πM

). This will mean that the minimum value of h(θ, πM

)

occurs either at θ = 0 or θ = γs. Notice that the function

m(x) = sec2 x tan−2s+2 x

has derivative

39

2 sec2 x tan−s−4s+2 x

s+ 2((s+ 1) tan2 x− 1),

which implies that h′(θ, πM

) is one-to-one on [0, γs]. Now if the minimum of h(θ, πM

)

occurs at γs, then convexity of gs (where gs is defined in the previous proposition)

concludes the proof. What remains therefore is to show that h(0, πM

) is positive.

We have that

h(

0,π

M

)=

1

2tan

ss+2

(π

M

)− tan

ss+2

(π

2M

).

This is positive precisely when

1

2tan

ss+2

(π

M

)> tan

ss+2

(π

2M

)

(1

2

) s+2s

tan(π

M

)> tan

(π

2M

)

(1

2

) s+2s

>tan

(π

2M

)tan

(πM

) .

and this inequality is satisfied for s sufficiently large, since the quantity on the right

increases to 12

as M increases to infinity (i.e. it is always less than 12).

We are now in a position to prove:

Theorem III.1.4. Let M ∈ N be given. Then there is an s0 = s0(M) so that for

all s > s0, the infimum in (25) is attained uniquely by the regular M -gon; that is,

fs(M) = Ma(M)s2 .

40

Proof. Fix M ∈ N. Proceeding as in the beginning of this chapter,

(M∑i=1

r−si

)(A(P ))s/2 ≥

(M∑i=1

r−si

)(M∑i=1

r2i tan θi

)s/2

where θi is the average of θ+i and θ−i .

=

(M∑i=1

r−si

)(M∑i=1

r2i tan θi

)s/2

=

( M∑i=1

r−si

) 2s+2

·(M∑i=1

r2i tan θi

) ss+2

s+2

2

.

Using Holder’s inequality,

( M∑i=1

r−si

) 2s+2

·(M∑i=1

r2i tan θi

) ss+2

s+2

2

≥(M∑i=1

(tan θi)ss+2

) s+22

. (26)

The idea now is to take advantage of the fact that the interval of concavity of

gs(x) = tanss+2 x is small when s is large together with the fact that the average of

the angles is π/M (and π/M is far from γs for s large) to show that even though gs

is not convex on (0, π/2), (17) still holds. Applying (17) to our current lower bound

in (26) will give

(M∑i=1

(tan θi)ss+2

) s+22

≥M1+s/2(tan π/M)s/2 = Ma(M)s/2.

Therefore our goal is to establish (17) for the sum

M∑i=1

tanss+2 θi.

Relabel the angles so that θ1 < θ2 < · · · < θM ; that is, the central angles of P

are now listed in increasing order. If θ1 ≥ γs, then all the angles are in the region of

41

convexity, and Jensen’s inequality applies. If θ1 < γs, then the previous lemma gives

that, for s sufficiently large,

tanss+2 θ1 + tan

ss+2 θM

2≥ tan

ss+2

(θ1 + θM

2

). (27)

Since θM must be at least the average πM

, this will be sufficient provided s is chosen

large enough such that it ensures that γs <π

2M. Indeed, if (27) holds, replace the

configuration (θ1, . . . , θM) by the configuration(θ1+θM

2, θ2, . . . , θM−1,

θ1+θM2

), which

decreases the quantity in (23); and if θ2 < γs, then there still must be another angle

that is greater than the average of πM

, so repeat the procedure above. This concludes

the proof.

Remark III.1.5. The quantity in (23) is not optimized by the regular M -gon

and central point for all values of s. To see this, notice that the M -gon is determined,

up to scaling, by the angles (θ−1 , θ+1 , . . . , θ

−M , θ

+M). Since the quantity in (23) is scale-

invariant, it is completely determined by these angles. In light of the relations

xi = ri sec θ−i = ri−1 sec θ+i−1,

if θ−i = θ+i for all i = 1, 2, . . . ,M , the quantity in (23) simplifies to

(M∑i=1

secs θi

)(M∑i=1

sin θi cos θi

) s2

, (28)

where θi := θ−i = θ+i . For M = 6 and s = 2, using the expression in (28) and the

configuration of angles (.809193, .0147173, .740205, .0241505, .774331, .778996) returns

a value of 20.089, which is lower than 6a(6) ≈ 20.785.

42

The next theorem gives a lower bound for the quantity in (23) that will be useful

in later sections.

Theorem III.1.6. For any M and any s ≥ 2, we have

fs(M) ≥ minν≤M

νa(ν)s2 .

Furthermore, this inequality is strict unless fs(M) = Ma(M)s2 .

Proof. As in Theorem III.1.4, the idea is to work with the expression

M∑i=1

tanss+2 θi

by arranging the central angles in increasing order. Specifically, consider the function

j(θ1, θM) = tanss+2 θ1 + tan

ss+2 θM

with the idea of holding the other angles fixed. This means that, for some k, the

above can be written as

j(θ1) = tanss+2 θ1 + tan

ss+2 (k − θ1).

The domain of this function is [0,min{γs, πM , k −πM}], with the added restriction

that k − θ1 <π2, since each central half-angle has measure at most π

2. Similar to the

function h in Theorem III.1.4, this function is first increasing because its derivative

is infinite at θ1 = 0. It has a unique maximum for the same reason that the function

h in Theorem III.1.4 has a unique maximum.

43

This means that the minimum of j occurs at either 0 or the right endpoint of its

domain of definition. If it occurs at the right endpoint, that means the minimum of

j occurs at either

• θ1 = γs, in which case convexity applies;

• θ1 = πM

; or

• θM = πM

.

If the minimum occurs at the left endpoint; that is, when θ1 = 0, then recast the

problem in terms of M − 1 angles and repeat the above procedure until the function

j has its minimum at the right endpoint of its domain for all the angles that have

measure less than πM

. This means that, for some n, either by convexity or by this

procedure one has that

(N∑i=1

tanss+2 θi

)1+ s2

≥ (M − n)a(M − n)s2 ≥ min

ν≤Mνa(ν)

s2 .

The remark about strict inequality in the statement of this theorem follows from

the technique just demonstrated together with the remarks at the beginning of this

chapter. More precisely, suppose that in the above technique, the angles θ1, . . . , θn

are replaced by 0, . . . , 0. (The other angles will also be reassigned in order to satisfy

the constraint.) Then for the polygon generated by these new central angles we have

that

(M∑i=1

r−si

)A(P )

s2 =

(n∑i=1

r−si

)A(P )

s2 +

M∑i=n+1

r−si

A(P )s2 ,

and using Holder’s inequality on the second term,

44

(n∑i=1

r−si

)A(P )

s2 +

M∑i=n+1

r−si

A(P )s2 ≥

(n∑i=1

r−si

)A(P )

s2 +

M∑i=n+1

tanss+2 θi

s+2

2

,

and this last quantity is strictly greater than the lower bound given in the statement

of the theorem.

III.2 The Constant Cs,d

The constant Cs,d appearing in the asymptotic result (12) of Hardin and Saff is worthy

of study because it is indicative of the local structure of s-optimal configurations when

s > d. The proof of Theorem III.2.3 will reflect this fact most concretely in the case

d = 2, but intuitively one can think of this as a reflection of the local character of

the Riesz kernel when s > d, since the first term in the asymptotic expansion should

reflect the nearest neighbor structure of an optimal configuration.

Since selecting any N -point configuration on a set A and computing its Riesz

s-energy gives an upper bound for Es(A,N), upper bounds for Cs,d can be readily

attained. In this section lattices will be used to give an upper bound for the constant

Cs,d defined by (12). The computation is instructive and provides some intuition

concerning the order of growth of Es(A,N) as N approaches infinity. This argument

can be found in [23] and references therein.

Consider any d-dimensional lattice L ⊆ Rd with fundamental cell F . Given N ,

shrink the lattice by a factor of 1/(N − 1):

1

N − 1L =

{1

N − 1x : x ∈ L

}.

45

Consider now ( 1N−1

L) ∩ F . Then it is easy to see that this set contains Nd points of

the entire scaled lattice. Call these Nd points ωNd and estimate the quantity

Es(ωNd)

(Nd)1+ sd

for a fixed s > d. The difficulty in computing the energy Es(ωNd) arises from the

different local structures of the individual points in the lattice restricted to the fun-

damental domain F . To alleviate this problem, estimate the energy for each point in

( 1N−1

L) ∩ F by estimating

∑x,y∈( 1

N−1L)∩F

x6=y

1

|x− y|s

from above by

∑x∈( 1

N−1L)∩F

∑

y∈ 1N−1

L

y 6=x

1

|x− y|s

.

It therefore follows that

Es(ωNd) =∑

x,y∈( 1N−1

L)∩Fx 6=y

1

|x− y|s≤

∑x∈( 1

N−1L)∩F

∑

y∈ 1N−1

L

y 6=x

1

|x− y|s

= NdζL(s)(N−1)s.

The factor (N − 1)s comes from the scaling of the lattice by the factor 1/(N − 1).

Hence

Es(ωNd)

(Nd)1+ sd≤ NdζL(s)(N − 1)s

Nd+s≤ ζL(s).

46

The existence of the limit in (12) now implies that

Cs,d = limN→∞

Es(F,N)

N1+ sd|F |

sd ≤ lim sup

N→∞

Es(ωNd)

(Nd)1+ sd|F |

sd ≤ ζL(s)|F |

sd , (29)

where |F | denotes the d-dimensional Lebesgue measure of F (in this case, equal to

the d-dimensional Hausdorff measure of F ).

The natural question that follows is to ask which lattice gives the best estimate

for Cs,d. When d = 1, the integer lattice Z gives the best bound, and in fact in ([25])

it was shown that

Cs,1 = 2ζ(s),

where ζ(s) denotes the classical Riemann zeta function. The value of Cs,d is not

known for any other values of d, but the results of Montgomery, Cohn, and Kumar

mentioned in section II.5 serve as a starting point in dimensions 2, 8, and 24.

The efforts made in this thesis have had as their objective progress toward a

computation of Cs,2. For this reason much of the work of the thesis concerns results

from plane geometry.

Denoting by Λ the hexagonal lattice in the plane as defined a section ago, in ([23])

it was conjectured that, for s > 2,

Cs,2 =

(√3

2

) s2

· ζΛ(s).

The result of (29) gives

Cs,2 ≤(√

3

2

) s2

· ζΛ(s). (30)

Before proceeding, we require the following lemmas:

47

Lemma III.2.1. Let {ωN} be a sequence of s-optimal configurations on the

unit square, where s > 2. Then the number of points whose Voronoi cell meets the

boundary of the unit square is o(N) as N →∞.

Proof. Let ε > 0 be a fixed small number. Let Sε denote the complement in the unit

square of the points inside the open square centered at (1/2, 1/2) with area 1 − ε2.

Let δ be the perpendicular distance from (1/2, 1) to the open square just described.

Choose m ∈ N so that m > 32/δ2. Partition then the upper strip of Sε into 4m

equal rectangles of height δ/4 and width 1m

. For N sufficiently large, each of the 4m

rectangles just constructed will have points from ωN in them, by (13). Notice then

that any point in the third row of these rectangles cannot have a Voronoi cell that

meets the upper boundary of the unit square, since the perpendicular bisector of any

point in the third row of rectangles and any point in the corresponding rectangle two

above it does not touch that boundary, by choice of m. Similarly for points in the

fourth row of rectangles and below, and similarly in turn for points in the other strips

of Sε.

This implies that all of the points that have a Voronoi cell meeting the boundary

of the unit square lie in the outer half of Sε. Since Theorem II.4.4 implies that

limN→∞

|Sε ∩ ωN |N

< ε,

and ε was arbitrarily small, we have the claim.

Remark III.2.2. Lemma III.2.1 also holds for compact sets A ⊆ R2 satisfying

the following condition: For every ε > 0, there is a polygon P ⊆ A whose area is

within ε of the area of A. The proof is similar and involves choosing a polygon as

just described, then fattening each edge into a small rectangle and proceeding as in

the lemma.

48

Lemma III.2.3.

lims→∞

ζΛ(s) = 6,

where Λ denotes the hexagonal lattice in R2 as defined in Chapter 2.

Proof. Let H0 denote the counting measure on R2. As all points in Λ− {0} are at a

distance of at least 1 from the origin, one has that

ζΛ(s) =∑y∈Λy 6=0

1

|y|s=∫

Λ−{0}

1

|y|sdH0(y),

and the integrand is monotonically decreasing with s. Therefore, by the monotone

convergence theorem,

lims→∞

ζΛ(s) = lims→∞

∫Λ−{0}

1

|y|sdH0(y) = 6,

as the quantity 1|y|s approaches 0 as s approaches infinity for all points at a distance

strictly greater than 1 from the origin.

We now have the following:

Theorem III.2.4. lims→∞

Cs,2(√3

2

) s2

= 6.

Proof. In light of the upper estimate for Cs,2 in (30), it follows from Lemma III.2.3

that

lims→∞

Cs,2(√3

2

) s2≤ 6.

49

Therefore, it is only necessary to show the reverse inequality.

By Theorem III.1.4, choose s big enough so that fs(M) is given by Ma(M)s2 for

M ≤ 7. Then (xa(x)s/2)2s+2 is convex and decreasing on [3, βs], where βs is the place

where fs takes its minimum. Let M ∈ N be the value such that Ma(M)s2 takes its

least value, and let N ′ be the number of points in an s-optimal configuration ωsN

on the unit square whose Voronoi cells do not meet the boundary, and let νi be the

number of edges of the i-th Voronoi cell. Relabel ωsN so that {ωsN}N′

i=1 is the collection

of points in ωsN whose Voronoi cells do not meet the boundary. Then by Theorem

III.1.6,

2sEs(ωsN) · A([0, 1]2)

s2 ≥

N∑i=1

νi∑j=1

r−sij

( N∑i=1

A(Vi)

) s2

≥

N ′∑i=1

fs(νi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

≥

N ′∑i=1

φs(νi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

, (31)

where

φs(x) =

xa(x)

s2 , x ≤M

Ma(M)s2 , x > M

.

Using Holder’s inequality in (31) gives

N ′∑i=1

φs(νi)

A(Vi)s2

2s+2

N ′∑i=1

A(Vi)

ss+2

s+2

2

≥

N ′∑i=1

φs(νi)2s+2

s+2

2

,

and by convexity, choice of s, and Lemmas II.6.4 and II.6.5, for N sufficiently large

one has

50

N ′∑i=1

φs(νi)2s+2

s+2

2

≥ (N ′)1+ s2φs

(∑N ′

i=1 νiN ′

)= (N ′)1+ s

2φs

(6N

N ′

)=

6N

N ′(N ′)1+ s

2a(

6N

N ′

) s2

.

In light of Lemma III.2.1,

limN→∞

N ′

N= 1.

Thus,

Cs,2 = limN→∞

Es([0, 1]2, N)

N1+ s2

≥ 6 · 2−sa(6)s2 = 6

(√3

2

) s2

,

whence the result follows.

Let Vi denote the i-th Voronoi cell. In what follows, we will set

∆s(Vi) :=

(νi∑i=1

r−si

)A(Vi)

s2 − φs(νi)

so that ∆s(Vi) gives a difference between the value of(∑νi

i=1 r−si

)A(Vi)

s/2 and the

lower bound used in Theorem III.2.4.

51

chapter iv

geometric constraints on certainconfigurations

IV.1 Some Results Concerning the Geometry of s-Optimal

Configurations for Large s

Given a compact set A ⊆ Rp, one can relate a portion of the energy of an N -point

configuration ωN to the structure of the Voronoi diagram. In this section, it will be

shown that, for large values of the exponent s, certain types of Voronoi cells associated

to s-optimal configurations will comprise a small proportion of the total number of

Voronoi cells. The aim of this strategy is then to estimate the energy from below by

considering only the nearest neighbors of points whose Voronoi cells are in some sense

desirable.

Results are first demonstrated for the unit square [0, 1]× [0, 1] in R2, and then for

a larger class of sets in the plane. Results for sets of Hausdorff dimension 2 embedded

in higher-dimensional sets are then discussed.

Let ωsN denote an s-optimal configuration in the unit square in R2, and fix γ ∈

(0, 1). It shall be shown that the points whose closest neighbor is at a distance of less

than δ(N) = 2√Na(6)

form a small proportion of the s-optimal configuration ωsN .

To this end, define

AsN,γ = {x ∈ ωsN : r(x) < γδ(N)},

where r(x) = min{|x− y| : y ∈ ωsN and y 6= x}. If y ∈ ωsN satisfies r(x) = |x− y|, we

shall call y a closest neighbor of x.

52

Let C > 6. Since, by Lemma III.2.3,

lims→∞

ζΛ(s) = 6,

there is s0 sufficiently large such that

ζΛ(s) < C,

for all s > s0.

Theorem IV.1.1. Let γ ∈ (0, 1), C > 6, and choose s0 as above. Then for all s > s0,

there is N0 = N0(s) such that for all N > N0,

|AsN,γ|N

≤ Cγs.

Proof. It was shown in an earlier section that

limN→∞

Es(ωsN)

N1+ s2≤ ζΛ(s)(

√3

2)s2 .

Thus, if s > s0, there is N0 such that for all N > N0,

Es(ωsN)

N1+ s2≤ C(

√3

2)s2 . (32)

To achieve a lower estimate, restrict the sum to only those points in AsN,γ and sum

only over one closest neighbor to each of those points, giving

Es(ωsN)

N1+ s2≥|AsN,γ|γ−s2−s(Na(6))

s2

N1+ s2

=|AsN,γ|N

γ−s(

√3

2)s2 , (33)

53

since a(6) = 2√

3. Combining (32) and (33), for all N > N0,

|AsN,γ|N

γ−s(

√3

2)s2 ≤ Es(ω

sN)

N1+ s2≤ C(

√3

2)s2 ,

whence

|AsN,γ|N

≤ Cγs.

Next the results from the previous section will be used to show that the proportion

of Voronoi cells that do not have precisely six sides is also small for large s.

Theorem IV.1.2 Let γ ∈ (0, 1) and C > 6. If BN,γ denotes the set of points

whose Voronoi cell has an inradius of at least γδ(N)2

but does not have exactly six

sides, then there is a positive constant D such that, for N = N(s) and s sufficiently

large,

|BN,γ|N

≤ D ·(

1

γ2− 1 + Cγs

).

(From now on, the dependence of sets such as BN,γ on s will be suppressed.)

Proof. Form a Voronoi decomposition for the configuration ωsN , but ignore the cells

whose centers are in the set AsN,γ. Then every Voronoi cell considered will have an

inradius of at least γδ(N)2

. Denote by V (x) the Voronoi cell centered at x by A(V (x))

the area of V (x). Then,

1 ≥∑

x∈ωsN\AsN,γ

A(V (x)) =∑

x∈GN,γA(V (x)) +

∑x∈BN,γ

A(V (x)), (34)

54

where GN,γ denotes those points x ∈ ωsN\AsN,γ whose Voronoi cell has exactly six

sides, and BN,γ denotes those points x ∈ ωsN\AsN,γ whose Voronoi cell does not have

exactly six sides. We have that

∑x∈GN,γ

A(V (x)) ≥ |GN,γ|(γδ(N)

2

)2

a(6). (35)

Set now

a(x) =

a(x), x ∈ [3, 5] ∪ [7,∞)

a(5) + (a(7)−a(5))2

(x− 5), x ∈ [5, 7].

Notice that a(x) is also convex and agrees with a(x) when x /∈ (5, 7). If V (x) is a

ν(x)-gon, it follows from Lemmas II.6.4 and II.6.5 that

∑x∈BN,γ

A(V (x)) ≥(γδ(N)

2

)2

|BN,γ| · a

∑x∈BN,γ

ν(x)

|BN,γ|

≥ (γδ(N)

2

)2

|BN,γ| · a(6)

(36)

=

(γδ(N)

2

)2

|BN,γ| · (a(6) + ∆a),

where ∆a = a(6)− a(6). Since |BN,γ|+ |GN,γ|+ |AsN,γ| = N , inserting (35) and (36)

into (40) gives

1 ≥(γδ(N)

2

)2 ((N − |AsN,γ|

)a(6) + |BN,γ|∆a

),

≥ γ2

(1− Cγs +

|BN,γ|N

∆a

a(6)

),

and one notices the use of Theorem IV.1.1, with C defined as in that theorem. Hence,

55

|BN,γ|N

≤(

1

γ2− 1 + Cγs

)(a(6)

∆a

).

What we have shown thus far is that a fraction, as close to 1 as desired, of all

the Voronoi cells of an s-optimal configuration are hexagons with at least a radius

γδ(N)/2 for sufficiently large N and s. The next logical result will be to form an

upper bound on the area on the Voronoi cells.

Theorem IV.1.3. Let Γ > 1, γ ∈ (0, 1) be fixed, and let DN,γ,Γ denote those points

in ωsN whose Voronoi cells have six sides, an inradius at least γδ(N)/2, and area

more than Γ/N . Then for any C > 6, there are N0 (again depending on s) and s0

sufficiently large such that, for N > N0 and s > s0,

|DN,γ,Γ|N

≤ 1− γ2 (1− Cγs)Γ− γ2

.

Proof. We proceed in a similar way to the previous theorem. Denote by V (x) the

Voronoi cell with center x ∈ ωsN , and by BN,γ those points in ωsN whose Voronoi cells

do not have six sides and do have inradius at least γδ(N)/2. Then one has

1 ≥∑

x∈BN,γA(V (x)) +

∑x∈GN,γ\DN,γ,Γ

A(V (x)) +∑

x∈DN,γ,ΓA(V (x))

≥(γδ(N)

2

)2|BN,γ|

∑x∈BN,γ

a(ν(x))

|BN,γ|+ |GN,γ\DN,γ,Γ| · a(6)

+ |DN,γ,Γ|Γ

N.

As in Theorem IV.1.2, the first term inside the large parentheses above is bounded

below by |BN,γ| · a(6). Hence, the above quantity is bounded below by

56

≥(γδ(N)

2

)2

(|BN,γ| · a(6) + |GN,γ\DN,γ,Γ| · a(6)) + |DN,γ,Γ|Γ

N

≥ γ2

(|BN,γ|N

+|GN,γ\DN,γ,Γ|

N

)+ |DN,γ,Γ|

Γ

N.

As N = |AsN,γ|+|BN,γ|+|GN,γ\DN,γ,Γ|+|DN,γ,Γ|, it follows from the previous theorems

of this section that, for N and s sufficiently large,

1 ≥ γ2 (1− Cγs) +|DN,γ,Γ|N

·(Γ− γ2

),

whence

|DN,γ,Γ|N

≤ 1− γ2 (1− Cγs)Γ− γ2

.

Again notice that the right side of this inequality can be made as small as desired

for any fixed Γ by appropriate choices of γ and s.

Remark IV.1.4. Suppose that {ωN} is a sequence of asymptotically uniformly

distributed N -point configurations on the unit square satisfying

Es(ωN)

N1+ s2≤ λ

(√3

2

) s2

for some λ > 6 and N sufficiently large. Then Theorems IV.1.1, IV.1.2, and IV.1.3

all hold with C replaced by λ. This remark extends analogously to the results in the

following two sections that are analogues of Theorems IV.1.1, IV.1.2, and IV.1.3.

57

IV.2 Analogous Results for S2

Only a small amount of effort is required to extend the results of the previous section

to the two-dimensional manifold S2 ⊆ R3. First, observe that in the formation of

Voronoi cells, the plane that perpendicularly bisects any pair of points will intersect

S2 in a great circle. Therefore, the Voronoi cells on S2 will be spherical polygons

rather than the Euclidean polygons seen in the previous section.

For simplicity, in what follows we will denote by S2 the sphere in R3 centered at

the origin and having unit area rather than unit radius. Also, it must be recalled

that the energy is calculated using Euclidean distance in all cases; geodesic distance

on the sphere is not used in the following calculations.

Let ωsN be an s-optimal configuration on S2. As in the previous, denote

AsN,γ = {x ∈ ωsN : r(x) < γδ(N)},

where r(x) = min{|x − y| : y ∈ ωsN and y 6= x}, and δ(N) = 2√Na(6)

. Then we have

the following:

Theorem IV.2.1. Let C > 6, and choose s0 so that ζΛ(s) < C for all s > s0. Then

if s > s0, there is N0 (depending on s) such that for all N > N0,

|AsN,γ|N

≤ Cγs.

Proof. The proof follows exactly the proof of Theorem IV.1.1. Summing only over

one closest neighbor of each point in AsN,γ, then for N sufficiently large,

Es(ωsN)

N1+ s2≥|AsN,γ|γ−s2−s(Na(6))

s2

N1+ s2

=|AsN,γ|N

γ−s(

√3

2)s2 .

58

Using again the upper estimate on Cs,2 from an earlier section and recalling that S2

has unit area, it follows that, for s sufficiently large,

|AsN,γ|N

γ−s(

√3

2)s2 ≤ Es(ω

sN)

N1+ s2≤ C(

√3

2)s2 ,

whence

|AsN,γ|N

≤ Cγs.

To extend the results of the previous section concerning Voronoi cells, we need

analogues of the lemmas due to Fejes-Toth. First note that Lemma II.6.5 holds also

for spherical graphs, since Euler’s formula holds for those graphs. Our analogue of

Lemma II.6.4 requires the computation of the area of a regular spherical n-gon of

spherical inradius a.

Lemma IV.2.2 The area of a regular spherical n-gon of spherical inradius a is given

by

αa(n) = R2[2n · cos−1

(cos

(a

R

)· sin

(π

n

))− (n− 2)π

],

where R is the radius of the sphere.

Proof. It is well-known that the area of a spherical n-gon is equal to the so-called

angular deficiency; that is, it is the difference between the sum of all the spherical

angles, denoted θ, and (n− 2)π, scaled appropriately:

Area = (θ − (n− 2)π)R2

59

The strategy then is to triangulate the regular spherical n-gon. Ratcliffe observed

in [32] that in a right spherical triangle, if one of the angles β, other than the right

angle, and the length of the adjacent leg a are known, then the third angle α is given

by

cosα = cos(a

R

)sin β.

Triangulating the n-gon gives us 2n right spherical triangles each having one angle

with measure π/n. Hence the third angle of each of the 2n spherical triangles is

α = cos−1(

cos(a

R

)sin

(π

n

)),

whence the result follows.

Again it is a basic computation as in Lemma II.6.4 to show that

αa(x) = 2xR2 · cos−1(

cos(a

R

)sin

(π

x

))

is a strictly convex function of x. Now let a(N) denote the radius of a spherical cap

formed by intersecting S2 with another sphere centered at a point on S2 with radius

γδ(N)2

. One now has the following:

Theorem IV.2.3 Let BN,γ ⊆ ωsN denote those points not in AsN,γ and not having a

six-sided Voronoi cell. Then for N = N(s) and s sufficiently large, there are positive

constants C and D such that

|BN,γ|N

≤ D

1

γ2(1−O

(1N

)) − 1 + Cγs

.

60

Proof. Let GN,γ denote those points in ωsN but neither in AN,γ nor BN,γ. First notice

that

1 ≥∑

x∈BN,γA(V (x)) +

∑x∈GN,γ

A(V (x)). (37)

Now proceed as in the proof of Theorem IV.1.2. Let

αa(x) =

αa(x), x ∈ [3, 5] ∪ [7,∞)

αa(5) + (αa(7)−αa(5))2

(x− 5), x ∈ [5, 7].

As before, this function is convex and decreasing on [3,∞). In [16] it was shown that

the spherical n-gon with minimum area and a fixed spherical inradius is the regular

spherical n-gon circumscribed about a spherical cap of that radius. Returning to (37)

and applying the previous results of this section,

1 ≥ |BN,γ| ·∑

x∈BN,γ

αa(N)(ν(x))

|BN,γ|+ |GN,γ| · αa(N)(6),

where ν(x) denotes the number of sides of V (x). The convexity of αa(N) gives

1 ≥ |BN,γ| · αa(N)

(∑x∈BN,γ ν(x)

|BN,γ|

)+ |GN,γ| · αa(N)(6)

≥ |BN,γ| · αa(N)(6) + |GN,γ| · αa(N)(6),

Since |AsN,γ|+ |BN,γ|+ |GN,γ| = N ,

1 ≥ |BN,γ| · αa(N)(6) + (N − |AsN,γ| − |BN,γ|) · αa(N)(6)

61

= |BN,γ| ·(αa(N)(6)− αa(N)(6)

)+ (N − |AsN,γ|) · α γδ(N)

2

(6). (38)

since a(N) is greater than the straight line distance γδ(N)2

. To understand better

the behavior of αa(N)(6) for large N , look at the power expansion of αa(N)(6) (as

a function of a(N)) about a = 0. The power series expansion of cos−1(x) about

x = sin(π/6) = 1/2 is

cos−1(x) =π

3− 2√

3(x− 1

2) +O(x2),

and the power series expansion of 12

cos(a) about a = 0 is

1

2cos(a) =

1

2− 1

4a2 +O(a4).

It follows that the power series expansion of αa(N)(6) about a = 0 is given by

αa(N)(6) = 2√

3a2 −O(a4).

Notice now that

α γδ(N)2

(6) =γ2

N

(1−O

(1

N

)).

This last result together with (38) now imply that

1 ≥ |BN,γ| ·(αa(N)(6)− αa(N)(6)

)+ γ2

(1− |AN,γ|

N

)(1−O

(1

N

)),

dividing and multiplying the first term on the right by α γδ(N)2

(6) it follows that

62

1 ≥ γ2 |BN,γ|N

(1−O

(1

N

))·

(αa(N)(6)− αa(N)(6)

)α γδ(N)

2

(6)+γ2

(1− |AN,γ|

N

)(1−O

(1

N

)),

whence Theorem IV.2.1 gives

1

γ2(1−O

(1N

)) ≥ |BN,γ|N

·

(αa(N)(6)− αa(N)(6)

)α γδ(N)

2

(6)+ 1− Cγs,

with C defined as in that theorem. Hence,

|BN,γ|N

≤α γδ(N)

2

(6)(αa(N)(6)− αa(N)(6)

) 1

γ2(1−O

(1N

)) − 1 + Cγs

.It must be mentioned that, as N approaches infinity, a computation shows that the

term outside parentheses approaches roughly 91.7, so it is finite for all N . This

concludes the proof.

The next result is the direct analogue of Theorem IV.2.3:

Theorem IV.2.4 Let Γ > 1 be fixed, and let DN,γ,Γ denote those points in GN,γ (as

defined in the previous theorem) whose Voronoi cells have area more than Γ/N . Then

for any C > 6, there are N0 (depending on s) and s0 sufficiently large such that, for

N > N0 and s > s0,

|DN,γ,Γ|N

≤1− γ2 (1− Cγs)

(1−O

(1N

))Γ− 1

Proof. Proceeding as in the proof of Theorem IV.1.3,

1 ≥∑

x∈BN,γA(V (x)) +

∑x∈GN,γ\DN,γ,Γ

A(V (x)) +∑

x∈DN,γ,ΓA(V (x))

63

≥ |BN,γ| · αa(6) + |GN,γ\DN,γ,Γ| · αa(6) +Γ

N|DN,γ,Γ|.

As |AsN,γ|+ |BN,γ|+ |GN,γ\DN,γ,Γ|+ |DN,γ,Γ| = N and a ≥ γδ(N)2

, it follows that

1 ≥(N − |AsN,γ| − |DN,γ,Γ|

)· α γδ(N)

2

(6) +Γ

N|DN,γ,Γ|,

whence

|DN,γ,Γ|N

≤1− α γδ(N)

2

(6) · (N − |AsN,γ|)Γ− α γδ(N)

2

(6)≤

1− γ2 (1− Cγs)(1−O

(1N

))Γ− 1

.

Again these results reflect that the sets AsN,γ, BN,γ, and DN,γ,Γ make up a small

proportion of all the points in ωsN . This gives very good constraints on the Voronoi

cells: An arbitrarily large fraction of them are hexagons of area arbitrarily close to

1/N (in the unit square and the sphere of unit area) with the appropriate choice of

s.

IV.3 Results in the Plane

The first and simplest extension of the theorems IV.1.1, IV.1.2, and IV.1.3 are to all

convex polygons of six or fewer sides, because the essential results of Fejes-Toth (see

[16]) that were used in Theorem IV.1.2 all apply to such polygons. The only subtlety

to which one must pay some attention is the area of the polygon, which requires only

a slight adjustment. That adjustment is to change the packing radius δ(N). For a

figure of unit area, the packing radius is δ(N) = 2√Na(6)

, but for figures P of arbitrary

64

area, the packing radius becomes δP (N) = 2(Area(P ))12√

Na(6). Indeed, here is how the results

are extended to these polygons:

Theorem IV.3.1 Let P be a convex polygon of six or fewer sides and area A(P ).

Define the sets AN,γ,s, BN,γ, and DN,γ,Γ as in section 1, but replacing δ(N) with δP (N)

where appropriate. Then the same inequalities given in Theorems IV.1.1, IV.1.2, and

IV.1.3 hold.

Proof. The proof follows the proof of Theorem IV.1.1. Given C > 6, s can be chosen

so that

ζΛ(s) < C.

It follows from previous results then that for N = N(s) sufficiently large,

Es(ωsN)

N1+ s2≤ C

(√3

2

) s2

· A(P )−s2 .

We achieve a lower bound for Es(ωsN) by summing over only one closest neighbor to

each point in AN,γ,s:

Es(ωsN) ≥ |AN,γ,s| · γ−s · 2−s · A(P )−s/2 · (Na(6))s/2,

and dividing now by N1+ s2 gives us that

|AN,γ,s| · γ−s · 2−s · A(P )−s/2 · (Na(6))s/2

N1+ s2

=|AN,γ,s| · γ−s · A(P )−s/2

N·(√

3

2

) s2

≤ Es(ωsN)

N1+ s2

65

≤ C

(√3

2

) s2

· A(P )−s2 ,

whence the first result follows. The other results have exactly the same proofs as

section IV.1, with a similar cancellation of the term A(P ).

An important and natural question to ask is to what class of sets A ⊆ R2 we

hope to be able to extend the results of section IV.1. First it is important to note

that an important tool in the proof of Theorem IV.1.1 is the asymptotic result of

Theorem 2.4 in [20] (Theorem II.4.4), which requires only that the set A be compact.

On the other hand, it would be convenient to restrict the Voronoi diagrams to the set

A itself and to make use of the concept of area. Therefore, the sets considered will

be connected and compact with nonempty interior. However, to extend the results

of section 1, it was also necessary to add a restriction on A that is something of a

rectifiability condition; this condition is precisely the condition given in the Remark

III.2.2. Namely, for every ε > 0, there must be a polygon P inside A such that

Area(A \ P ) < ε.

Admitting these conditions gives the following:

Theorem IV.3.2 Let K be a compact set of area A(K) satisfying the rectifiability

condition above and ε > 0, γ ∈ (0, 1) be given. Let C > 6 be given, choose s as

in Theorem IV.1.1, and let ωsN be an s-optimal configuration on K. Then given the

usual definition of the set

AN,γ,s = {x ∈ ωsN : r(x) < γδK(N)},

66

for all N = N(s) sufficiently large we have

|AN,γ,s|N

< Cγs.

Proof. The proof is exactly the same as the proof of Theorem IV.3.1 shown above.

Theorem IV.3.3 Under the same conditions of Theorem IV.3.2, given ε > 0, and

given the usual definition of the set

BN,γ = {x ∈ ωsN\AN,γ,s : The Voronoi cell of x is not a 6-sided polygon},

there is N = N(s, ε) sufficiently large and γ ∈ (0, 1) so that

|BN,γ|N

< ε.

Proof. For any ε > 0, by hypothesis one construct a polygon P in the interior of K

whose area is within ε of the area of K. Given this polygon, it can then be decomposed

it into a union of triangles. Define now Fρ to be those points in K that are within

a distance ρ of the boundary of one of these triangles or outside of the polygon P .

Given now ε > 0, choose P , corresponding triangles {Ti}mi=1, and ρ such that

Area(Fρ) <ε · Area(K)

4.

By Theorem II.4.4, for all N sufficiently large, the number of points in ωsN that are

in Fρ is no more than εN2

. In addition, choose N sufficiently large so that those points

not in Fρ do not have Voronoi cells that would gain an edge by the superposition of

the triangles on the Voronoi diagram (here again Lemma III.2.2 is being used). Then

67

A(K) ≥∑

x∈ωsN\AN,γ,s

A(V (x)) =∑

x∈BN,γ∩FρA(V (x))+

∑x∈BN,γ\Fρ

A(V (x))+∑

x∈GN,γA(V (x)),

where GN,γ is the set of those points not in AN,γ,s∪Fρ whose Voronoi cell is a hexagon.

Now work in each triangle Ti individually, proceeding as in Theorem IV.1.2 above:

A(Ti) ≥∑

x∈GN,γ∩TiA(V (x)) +

∑x∈(BN,γ\Fρ)∩Ti

A(V (x))

≥ |GN,γ∩Ti|(γδK(N)

2

)2

·a(6)+

(γδK(N)

2

)2

|(BN,γ\Fρ)∩Ti|·a

∑x∈(BN,γ\Fρ)∩Ti

ν(x)

|(BN,γ\Fρ) ∩ Ti|

,where a(x) is the convex, decreasing function defined in Theorem IV.1.2. Again

proceeding as in that theorem,

a

∑x∈(BN,γ\Fρ)∩Ti

ν(x)

|(BN,γ\Fρ) ∩ Ti|

≥ a(6),

whence

A(Ti) ≥(γδK(N)

2

)2

[|GN,γ ∩ Ti| · a(6) + |(BN,γ\Fρ) ∩ Ti| · a(6)] .

Summing both sides over i,

A(K) ≥ A(P ) ≥(γ2A(K)

Na(6)

)(|GN,γ| · a(6) + |BN,γ\Fρ| · a(6)) .

as N = |AN,γ,s|+ |BN,γ\Fρ|+ |Fρ|+ |GN,γ|,

A(K) ≥(γ2A(K)

Na(6)

)((N − |AN,γ,s| − |BN,γ\Fρ| − |Fρ|) · a(6) + |BN,γ\Fρ| · a(6)

)

68

=(γ2A(K)

)(1− |AN,γ,s|

N− |Fρ|

N+

(|BN,γ\Fρ|

N

)(a(6)− a(6)

a(6)

)),

whence

|BN,γ\Fρ|N

≤ D ·(

1

γ2− 1 + Cγs +

|Fρ|N

),

which can be made arbitrarily small for appropriate choices of s and γ, and since

|Fρ|N

< ε2, giving the result.

The next theorem is the natural extension of Theorem IV.1.3.

Theorem IV.3.4 Let Γ > 1, ε > 0 be fixed, and let DN,γ,Γ denote those points in

GN,γ (as defined in the previous theorem) whose Voronoi cells have area more than

Γ·A(K)N

. Then for all s and N = N(s) sufficiently large, there is γ sufficiently close to

1 so that

|DN,γ,Γ|N

< ε.

Proof. The proof again uses a decomposition of the region as in Theorem IV.3.3, and

again the fact that one can choose a set arbitrarily small measure inside which only

a small proportion of the minimal energy points can lie, proceeding as in the proof of

Theorem IV.1.3.

IV.4 Holder Inequality Techniques and Refined Estimates

Given a compact set A ⊆ R2, some of the upper bounds for certain classes of Voronoi

cells given in a previous chapter can be improved by using lower estimates of the en-

69

ergy given by considering only the nearest neighbors of points away from the bound-

ary of A. In addition, similar techniques give an improvement on estimates for the

quantity Cs,2 described in an earlier chapter.

Lemma IV.4.1. Let a(x) = x tan πx, which is a decreasing, convex function

on [3,∞) satisfying limx→∞ a(x) = π. Then, furthermore, a(x)ss+2 is convex and

decreasing on [3,∞) for s ≥ 2.

Proof. a′(x) = tan πx− π

xsec2 π

x, and

a′′(x) = 2π2

x3 sec2 πx

tan πx. In more generality, if b(x) = a(x)

ss+2 ,

b′(x) =s

s+ 2a(x)

−2s+2 · a′(x).

This is negative on [3,∞) since a′(x) is negative on [3,∞). In addition,

b′′(x) =−2s

(s+ 2)2a(x)

−s−4s+2 · (a′(x))2 +

s

s+ 2a(x)

−2s+2 · a′′(x).

=s

s+ 2a(x)

−s−4s+2

(a(x)a′′(x)− 2

s+ 2(a′(x))2

).

The quantity outside the parentheses above is always positive. Therefore, b(x)

will be convex if the quantity inside parentheses is positive. Notice that the quantity

inside parentheses is increasing as s increases. Therefore, it suffices to check that the

quantity inside parentheses is positive for all x in [3,∞) when s = 2.

To that end, it suffices to verify that

(x tan

π

x

)(2π2

x3sec2 π

xtan

π

x

)− 1

2

(tan

π

x− π

xsec2 π

x

)2

≥ 0.

This will hold precisely when

70

(x tan

π

x

)(2π2

x3sec2 π

xtan

π

x

)≥ 1

2

(tan

π

x− π

xsec2 π

x

)2

,

2π

x

(sec

π

xtan

π

x

)≥ π

xsec2 π

x− tan

π

x.

Notice the reversal of signs on the right side since a′(x) is negative, and a posi-

tive quantity belongs on the right. The above inequality leads to the corresponding

inequality

2π secπ

xtan

π

x+ x tan

π

x≥ π sec2 π

x,

which will certainly hold if

2π secπ

xtan

π

x+ π ≥ π sec2 π

x,

since x tan πx

= a(x) ≥ π. Now multiplying on both sides by cos2 πx/π gives

2 sinπ

x+ cos2 π

x≥ 1.

This inequality is certainly true, since 2 sin πx≥ sin π

x≥ sin2 π

x.

Theorem IV.4.2. Let C > 6. Choose s0 so that ζΛ(s0) < C. Let B6 ⊆ ωsN denote

the set of points in the unit square whose Voronoi cell does not have six sides and does

not meet the boundary of the unit square. Then for every s > s0, there is N0 = N0(s)

and a constant D such that for all N > N0,

|B6|N≤ D

(C

2s+2 − 1

).

71

Proof. Let ξsN ⊆ ωsN be those points whose Voronoi cells do not meet the boundary

of the unit square. Say that |ξsN | = N ′. For a point xi ∈ ξsN , let {rij}νij=1 denote the

collection of perpendicular distances from xi to the νi edges of its Voronoi cell. Then

Es([0, 1]2, N)(A([0, 1]2) ≥ 2−s

N ′∑i=1

νi∑j=1

r−sij

N ′∑i=1

A(Vi)

s2

,

where A(Vi) denotes the area of the i-th Voronoi cell. For convenience, set R−si =∑νij=1 r

−sij . It follows then that

Es([0, 1]2, N)(A([0, 1]2) ≥ 2−s

N ′∑i=1

R−si

N ′∑i=1

A(Vi)

s2

= 2−s

N ′∑i=1

R−si

2s+2

N ′∑i=1

A(Vi)

ss+2

s+2

2

.

Applying Holder’s inequality then gives

2−s

N ′∑i=1

R−si

2s+2

N ′∑i=1

A(Vi)

ss+2

s+2

2

≥ 2−s

N ′∑i=1

(R−2i A(Vi)

) ss+2

s+2

2

.

Let now ri = minj rij. In light of the obvious inequalities

R−2i ≥ r−2

i

A(Vi) ≥ r2i a(νi),

it follows that

2−s

N ′∑i=1

(R−2i A(Vi)

) ss+2

s+2

2

≥ 2−s

N ′∑i=1

a(νi)ss+2

s+2

2

. (39)

72

By the previous lemma, a(x)ss+2 is convex and decreasing for all s ≥ 2. Set now

a(x) =

a(x)

ss+2 , x ∈ [3, 5] ∪ [7,∞)

a(5)ss+2 + (a(7)

ss+2−a(5)

ss+2 )

2(x− 5), x ∈ [5, 7].

Notice that a is also convex and decreasing on [3,∞). Let now N6 = N ′ − |B6|.

Picking up from (39), notice that

2−s

N ′∑i=1

a(νi)ss+2

s+2

2

= 2−s

N6a(6)ss+2 +

∑xi∈B6

a(νi)ss+2

s+2

2

.

The convexity of a and Fejes-Toth’s result now give us a lower bound:

2−s

N6a(6)ss+2 +

∑xi∈B6

a(νi)ss+2

s+2

2

≥ 2−s

N6a(6)ss+2 + (N ′ −N6)a

(6(N −N6)

N ′ −N6

) ss+2

s+2

2

= 2−s

N ′a(6)ss+2 + (N ′ −N6)(a

(6(N −N6)

N ′ −N6

) ss+2

− a(6)ss+2 )

s+2

2

.

(Notice that a(

6(N−N6)N ′−N6

)approaches a(6) as N approaches ∞, by Lemma III.2.1 of

the previous chapter.) The upper bound for Cs,2 given earlier together with these

inequalities now imply that, for N sufficiently large,

Ca(6)s2N1+ s

2 ≥ Es([0, 1]2, N) ≥ (N ′)1+ s2a(6)

s2

(1 + (1− N6

N ′)∆s)

) s+22

, (40)

where ∆s = ∆s(N) =a

(6(N−N6)

N′−N6

) ss+2−a(6)

ss+2

a(6)ss+2

. Proceeding from (40) now gives that

73

1 + (1− N6

N ′)∆s ≤ C

2s+2

(N

N ′

),

whence

1− N6

N ′≤ D

(C

2s+2

(N

N ′

)− 1

),

where D = 1∆s

. Hence

|B6|N≤ D

(C

2s+2 − N ′

N

)− N

N ′+ 1 ≤ D′

(C

2s+2 − 1

),

for a suitable choice of a slightly-adjusted D′.

Computations of the quantity ∆s show that, for N sufficiently large, the constant

D in the theorem above can always be chosen (regardless of s) to be less than 100.

Notice that the upper bound on |B6|/N given here has no dependence on the inra-

dius of any of the cells, a difference from Theorem IV.1.2. This result can also be

generalized from the unit square to compact sets in the plane:

Corollary IV.4.3. Given C > 6, choose s0 as in Theorem IV.1.1. Let A ⊆ R2 be

a compact set such that for any ε > 0, there is a polygon P ⊆ A whose area is within

ε of the area of A. Then with B6 as defined in Theorem IV.4.2, for every s > s0 there

is N0 = N0(s) and a constant D so that for all N > N0,

|B6|N≤ D

(C

2s+2 − 1

).

Proof. Given ε > 0, choose a polygon P ⊆ A whose area is within ε of the area of

A. By Lemma III.2.1 of the previous chapter and Remark III.2.2, the number of

74

Voronoi cells that meet the boundary is o(N) as N →∞. Therefore, one can proceed

precisely as in Theorem IV.4.2 to gain the desired result.

Recall that in Chapter 3, we defined

∆s(Vi) :=

(νi∑i=1

r−si

)A(Vi)

s2 − φs(νi)

Then proceeding as in Theorem III.2.4, for any N -point configuration ωN on the unit

square,

Es(ωN) · A([0, 1]2)s2 ≥ 2−s

N ′∑i=1

φs(νi) + ∆s(Vi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

= 2−s

N ′∑i=1

φs(νi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

+ 2−s

N ′∑i=1

∆s(Vi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

. (41)

This motivates the following:

Theorem IV.4.4. Let {ωsN} be a sequence of s-optimal configurations on the unit

square for s > 2. Then

lim supN→∞

1

N

N ′∑i=1

∆s(Vi)2s+2

≤ (ζΛ(s)− 6)2s+2 · a(6)

ss+2 .

Proof. Let ωsN be an s-optimal configuration, and s > 2. Proceeding from (41),

applying Holder’s inequality to the last term gives

2−s

N ′∑i=1

∆s(Vi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

≥ 2−s

N ′∑i=1

∆s(Vi)2s+2

s+2

2

.

75

Applying Holder’s inequality followed by Jensen’s inequality on the other term in

the last expression of (41) gives the expression of Theorem III.2.4:

2−s

N ′∑i=1

φs(νi)

A(Vi)s2

N ′∑i=1

A(Vi)

s2

≥ 2−s(N ′)1+ s2φs

(6N

N ′

)= 2−s

6N

N ′(N ′)1+ s

2a(

6N

N ′

) s2

.

Thus,

Es(ωsN) · A([0, 1]2)

s2 ≥ 2−s

6N

N ′(N ′)1+ s

2a(

6N

N ′

) s2

+ 2−s

N ′∑i=1

∆s(Vi)2s+2

s+2

2

.

Let Λ again denote the hexagonal lattice in the plane. Then recalling that, by the

optimality of ωsN ,

limN→∞

Es(ωsN)

N1+ s2≤ ζΛ(s) ·

(√3

2

) s2

,

we have that

6N

N ′

(N ′

N

)1+ s2

a(

6N

N ′

) s2

+

1

N

N ′∑i=1

∆s(Vi)2s+2

s+2

2

≤ ζΛ(s) · a(6)s2 +Os(N),

where Os(N) indicates an expression that is O(N) but with some dependence on s.

Letting N tend to infinity gives

lim supN→∞

1

N

N ′∑i=1

∆s(Vi)2s+2

s+2

2

≤ (ζΛ(s)− 6) · a(6)s2 ,

whence

76

lim supN→∞

1

N

N ′∑i=1

∆s(Vi)2s+2

≤ (ζΛ(s)− 6)2s+2 · a(6)

ss+2 .

Recall that Theorem III.1.4 and Theorem III.1.6 state that if the lower bound

given by the function φ(νi) is attained, then it is uniquely attained by the regular

νi-gon. Therefore, one can interpret the above result as a sense in which the Voronoi

cells “approach” regular polygons. To be a bit more precise, for a given Voronoi cell

Vi, let us now analyze the quantity ∆s(Vi). Let ri,min be the minimum altitude length

drawn from the fixed point p, and let βi = A(Vi)r2i,mina(νi)

. Then we have

(νi∑i=1

r−si

)(A(Vi))

s2 ≥ r−si,min · β

s2i r

si,mina(νi)

s2 = β

s2i a(νi)

s2 .

Thus, we have

∆s(Vi)2s+2 ≥

(βs2i − νi

) 2s+2

+· a(νi)

ss+2 ,

where (·)+ denotes the positive part of (·). Theorem IV.4.4 now gives

lim supN→∞

1

N

N ′∑i=1

(βs2i − νi

) 2s+2

+· a(νi)

ss+2 ≤ (ζΛ(s)− 6) · a(6)

s2 . (42)

Again relabel the points so that {ωsN}N′′

i=1 is the set of all points in ωsN whose Voronoi

cell does not meet the boundary and is a hexagon. Then considering only those points

in (42) whose Voronoi cell is a hexagon gives

lim supN→∞

1

N

N ′′∑i=1

(βs2i − 6

) 2s+2

+· a(6)

ss+2 ≤ (ζΛ(s)− 6)

2s+2 · a(6)

ss+2 ,

whence

77

lim supN→∞

1

N

N ′′∑i=1

(βs2i − 6

) 2s+2

+≤ (ζΛ(s)− 6)

2s+2 ,

which can be viewed as another geometric constraint.

78

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